# Nonempty Tree Qualification, Paths, Leaves, and New-Edge Cycles

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261012-advance-20261017-r06-tree-foundations · 2026-10-12 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(0, [], [], Yes, No, No, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], [], Yes, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0001/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 2. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(1, [], [[0]], Yes, No, Yes, Yes)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], [[0]], Yes, No, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0002/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 3. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(4, [[1, 2], [1, 3], [2, 3]], [[0], [1, 2, 3]], No, Yes, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], [[0], [1, 2, 3]], No, Yes, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0003/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 4. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(4, [[0, 1], [2, 3]], [[0, 1], [2, 3]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [2, 3]], [[0, 1], [2, 3]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0004/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 5. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(4, [[0, 1], [1, 2], [2, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [1, 2], [2, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0005/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 6. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(4, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0006/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 7. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1]], [[0, 1], [2], [3], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1]], [[0, 1], [2], [3], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0007/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 8. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 2]], [[0, 2], [1], [3], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2]], [[0, 2], [1], [3], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0008/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 9. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [0, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0009/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 10. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 3]], [[0, 3], [1], [2], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3]], [[0, 3], [1], [2], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0010/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 11. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0011/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 12. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 2], [0, 3]], [[0, 2, 3], [1], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3]], [[0, 2, 3], [1], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0012/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 13. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0013/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 14. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 4]], [[0, 4], [1], [2], [3]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 4]], [[0, 4], [1], [2], [3]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0014/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 15. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [0, 4]], [[0, 1, 4], [2], [3]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 4]], [[0, 1, 4], [2], [3]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0015/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 16. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 2], [0, 4]], [[0, 2, 4], [1], [3]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 4]], [[0, 2, 4], [1], [3]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0016/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 17. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [0, 2], [0, 4]], [[0, 1, 2, 4], [3]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 4]], [[0, 1, 2, 4], [3]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0017/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 18. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 3], [0, 4]], [[0, 3, 4], [1], [2]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3], [0, 4]], [[0, 3, 4], [1], [2]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0018/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 19. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [0, 3], [0, 4]], [[0, 1, 3, 4], [2]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3], [0, 4]], [[0, 1, 3, 4], [2]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0019/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 20. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 2], [0, 3], [0, 4]], [[0, 2, 3, 4], [1]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3], [0, 4]], [[0, 2, 3, 4], [1]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0020/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 21. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[0, 1, 2, 3, 4]], Yes, No, Yes, Yes)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[0, 1, 2, 3, 4]], Yes, No, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0021/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 22. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[1, 2]], [[0], [1, 2], [3], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[1, 2]], [[0], [1, 2], [3], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0022/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 23. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0023/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 24. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0024/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 25. Components, Cycles, and Nonempty Tree Qualification

Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.

View answer and explanation(5, [[0, 1], [0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, Yes, Yes, No)

Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, Yes, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0025/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-component-qualification · candidate





## 26. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=-1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(0, [], No, [No, No], not-a-tree, not-evaluated)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, [No, No], not-a-tree, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0026/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 27. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(1, [], Yes, [Yes, Yes], [0], 0)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, [Yes, Yes], [0], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0027/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 28. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], not-a-tree, not-evaluated)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], not-a-tree, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0028/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 29. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=-1, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(3, [[0, 1], [1, 2]], Yes, [No, Yes], outside-vertex-set, not-evaluated)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [No, Yes], outside-vertex-set, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0029/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 30. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], [1], 0)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], [1], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0030/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 31. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(4, [[0, 1], [0, 2], [0, 3]], Yes, [Yes, Yes], [1, 0, 3], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], Yes, [Yes, Yes], [1, 0, 3], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0031/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 32. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], [0, 3], 1)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], [0, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0032/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 33. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [1, 0, 4], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [1, 0, 4], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0033/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 34. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [2, 0, 5], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [2, 0, 5], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0034/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 35. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [3, 0], 1)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [3, 0], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0035/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 36. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], [4, 1], 1)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], [4, 1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0036/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 37. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], [5, 0, 2], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], [5, 0, 2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0037/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 38. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [0, 1, 3], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [0, 1, 3], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0038/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 39. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 0, 4], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 0, 4], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0039/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 40. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 1, 0, 5], 3)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 1, 0, 5], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0040/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 41. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 1, 0], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 1, 0], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0041/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 42. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], [4, 0, 1], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], [4, 0, 1], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0042/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 43. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], [5, 2], 1)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], [5, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0043/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 44. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 2, 3], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 2, 3], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0044/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 45. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 2, 0, 4], 3)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 2, 0, 4], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0045/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 46. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 0, 5], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 0, 5], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0046/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 47. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 2, 0], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 2, 0], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0047/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 48. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], [4, 1], 1)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], [4, 1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0048/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 49. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], [5, 0, 2], 2)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], [5, 0, 2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0049/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 50. Original-Tree and Endpoint Guards before a Unique Path

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.

View answer and explanation(6, [[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 3], 1)

Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0050/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-unique-path-gate · candidate





## 51. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(0, [], No, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0051/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 52. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(1, [], Yes, Yes, 0, No, not-applicable, not-applicable, not-evaluated)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, Yes, 0, No, not-applicable, not-applicable, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0052/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 53. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(2)={0,…,1} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=1. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(2, [[0, 1]], Yes, Yes, 1, Yes, [0], [], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [[0, 1]], Yes, Yes, 1, Yes, [0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0053/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 54. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(4, [[1, 2], [1, 3], [2, 3]], No, Yes, 0, No, not-applicable, not-applicable, not-evaluated)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], No, Yes, 0, No, not-applicable, not-applicable, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0054/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 55. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=1. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(3, [[0, 1], [1, 2]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0055/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 56. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=3. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(3, [[0, 1], [1, 2]], Yes, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0056/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 57. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=2. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(4, [[0, 1], [0, 2], [0, 3]], Yes, Yes, 1, Yes, [0, 1, 3], [[0, 1], [0, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], Yes, Yes, 1, Yes, [0, 1, 3], [[0, 1], [0, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0057/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 58. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [0, 4]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [0, 4]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0058/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 59. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0059/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 60. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 2]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 2]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0060/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 61. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0061/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 62. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 4]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 4]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0062/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 63. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0063/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 64. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0064/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 65. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [1, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0065/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 66. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [2, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [2, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0066/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 67. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0067/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 68. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 4]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 4]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0068/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 69. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0069/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 70. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [2, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [2, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0070/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 71. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [1, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [1, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0071/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 72. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [2, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [2, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0072/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 73. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0073/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 74. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 4], [2, 3]], Yes)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 4], [2, 3]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0074/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 75. Degree-One Leaf Deletion without Renumbering

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)

A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0075/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-leaf-deletion · candidate





## 76. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(0, [], No, [No, No], Yes, No, No, not-applicable, not-applicable)

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, [No, No], Yes, No, No, not-applicable, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0076/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 77. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 0]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(1, [], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable)

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0077/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 78. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(2)={0,…,1} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(2, [[0, 1]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable)

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [[0, 1]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0078/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 79. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[2, 0]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [0, 1, 2], [0, 1, 2, 0])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [0, 1, 2], [0, 1, 2, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0079/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 80. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[-1, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(3, [[0, 1], [1, 2]], Yes, [No, Yes], Yes, No, No, not-applicable, not-applicable)

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [No, Yes], Yes, No, No, not-applicable, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0080/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 81. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[1, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable)

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0081/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 82. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], Yes, No, No, not-applicable, not-applicable)

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], Yes, No, No, not-applicable, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0082/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 83. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0083/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 84. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0084/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 85. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0085/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 86. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0086/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 87. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0087/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 88. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0088/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 89. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0089/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 90. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0090/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 91. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0091/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 92. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 3, 1, 0, 5], [2, 3, 1, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 3, 1, 0, 5], [2, 3, 1, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0092/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 93. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 4, 0, 5], [2, 4, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 4, 0, 5], [2, 4, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0093/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 94. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable)

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0094/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 95. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0095/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 96. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0096/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 97. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0097/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 98. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0098/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 99. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0099/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate





## 100. Qualified Tree Nonedge and Its Unique Cycle Witness

Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

View answer and explanation(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])

Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r06-tree-0100/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-density17-tree-add-nonedge-cycle · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261017-r06-tree-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261017-r06-tree-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261017-r06-tree-foundations/validation-report.json)