# Nonempty Finite Tree Foundations

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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-20261007-advance-20261009-r02-tree-foundations · 2026-10-07 · 100 questions

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



## 1. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0001/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 2. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0002/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 3. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0003/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 4. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0004/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 5. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0005/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 6. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0006/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 7. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2] and E=[[0, 1], [1, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0007/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 8. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2] and E=[[0, 1], [1, 2], [0, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0008/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 9. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0009/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 10. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0010/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 11. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0011/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 12. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0012/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 13. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0013/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 14. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [0, 2], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0014/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 15. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 2], [2, 3], [3, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0015/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 16. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2], [2, 3], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0016/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 17. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2], [2, 0], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0017/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 18. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0018/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 19. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0019/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 20. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[0, 1], [1, 2], [2, 3], [3, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0020/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 21. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[0, 1], [0, 2], [0, 3], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0021/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 22. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0022/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 23. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0023/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 24. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationYes

Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0024/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 25. Nonempty Finite-Graph Tree Eligibility

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?

View answer and explanationNo

Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0025/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-property · candidate





## 26. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 0 as an ordered vertex list, without repeating vertices.

View answer and explanation[0]

Follow the tree path in traversal order to obtain [0], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0026/)

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





## 27. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 1 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 1]

Follow the tree path in traversal order to obtain [0, 1], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0027/)

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





## 28. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2] and E=[[1, 0], [0, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 2 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 2]

Follow the tree path in traversal order to obtain [0, 2], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0028/)

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





## 29. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 3 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 3]

Follow the tree path in traversal order to obtain [0, 3], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0029/)

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





## 30. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 3 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 3]

Follow the tree path in traversal order to obtain [0, 3], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0030/)

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





## 31. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 4 to 4 as an ordered vertex list, without repeating vertices.

View answer and explanation[4]

Follow the tree path in traversal order to obtain [4], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0031/)

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





## 32. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 4 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 4]

Follow the tree path in traversal order to obtain [0, 4], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0032/)

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





## 33. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 4 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 4]

Follow the tree path in traversal order to obtain [0, 4], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0033/)

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





## 34. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[1, 0], [2, 0], [3, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 5]

Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0034/)

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





## 35. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 1], [1, 0], [3, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 5]

Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0035/)

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





## 36. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 1], [3, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 5 to 5 as an ordered vertex list, without repeating vertices.

View answer and explanation[5]

Follow the tree path in traversal order to obtain [5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0036/)

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





## 37. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 5]

Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0037/)

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





## 38. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 4], [3, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 5]

Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0038/)

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





## 39. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[4, 3], [3, 2], [2, 1], [1, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 5]

Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0039/)

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





## 40. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0040/)

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





## 41. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 1], [1, 0], [3, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 6 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[6]

Follow the tree path in traversal order to obtain [6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0041/)

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





## 42. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 1], [3, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0042/)

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





## 43. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0043/)

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





## 44. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 4], [3, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0044/)

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





## 45. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [2, 1], [4, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0045/)

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





## 46. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [4, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 6 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[6]

Follow the tree path in traversal order to obtain [6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0046/)

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





## 47. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[4, 3], [3, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0047/)

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





## 48. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 5], [4, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0048/)

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





## 49. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 6], [3, 5], [4, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0049/)

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





## 50. Unique Ordered Path in a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[5, 4], [4, 3], [3, 2], [2, 1], [1, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.

View answer and explanation[0, 6]

Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0050/)

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





## 51. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[]

Count normalized incident edges and select degree exactly one, obtaining []. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0051/)

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





## 52. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[0, 1]

Count normalized incident edges and select degree exactly one, obtaining [0, 1]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0052/)

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





## 53. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2] and E=[[1, 0], [0, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[1, 2]

Count normalized incident edges and select degree exactly one, obtaining [1, 2]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0053/)

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





## 54. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[1, 2, 3]

Count normalized incident edges and select degree exactly one, obtaining [1, 2, 3]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0054/)

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





## 55. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3]

Count normalized incident edges and select degree exactly one, obtaining [2, 3]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0055/)

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





## 56. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[1, 2, 3, 4]

Count normalized incident edges and select degree exactly one, obtaining [1, 2, 3, 4]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0056/)

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





## 57. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3, 4]

Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0057/)

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





## 58. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[3, 4]

Count normalized incident edges and select degree exactly one, obtaining [3, 4]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0058/)

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





## 59. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[1, 0], [2, 0], [3, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[1, 2, 3, 4, 5]

Count normalized incident edges and select degree exactly one, obtaining [1, 2, 3, 4, 5]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0059/)

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





## 60. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 1], [1, 0], [3, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3, 4, 5]

Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4, 5]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0060/)

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





## 61. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 1], [3, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3, 4, 5]

Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4, 5]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0061/)

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





## 62. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[3, 4, 5]

Count normalized incident edges and select degree exactly one, obtaining [3, 4, 5]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0062/)

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





## 63. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 4], [3, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3, 5]

Count normalized incident edges and select degree exactly one, obtaining [2, 3, 5]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0063/)

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





## 64. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[4, 3], [3, 2], [2, 1], [1, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[4, 5]

Count normalized incident edges and select degree exactly one, obtaining [4, 5]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0064/)

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





## 65. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[1, 2, 3, 4, 5, 6]

Count normalized incident edges and select degree exactly one, obtaining [1, 2, 3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0065/)

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





## 66. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 1], [1, 0], [3, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3, 4, 5, 6]

Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0066/)

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





## 67. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 1], [3, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3, 4, 5, 6]

Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0067/)

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





## 68. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[3, 4, 5, 6]

Count normalized incident edges and select degree exactly one, obtaining [3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0068/)

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





## 69. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 4], [3, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3, 5, 6]

Count normalized incident edges and select degree exactly one, obtaining [2, 3, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0069/)

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





## 70. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [2, 1], [4, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[3, 4, 5, 6]

Count normalized incident edges and select degree exactly one, obtaining [3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0070/)

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





## 71. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [4, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[3, 4, 5, 6]

Count normalized incident edges and select degree exactly one, obtaining [3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0071/)

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





## 72. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[4, 3], [3, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[4, 5, 6]

Count normalized incident edges and select degree exactly one, obtaining [4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0072/)

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





## 73. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 5], [4, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[3, 4, 6]

Count normalized incident edges and select degree exactly one, obtaining [3, 4, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0073/)

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





## 74. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 6], [3, 5], [4, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[2, 3, 4]

Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0074/)

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





## 75. Degree-One Leaves of a Finite Tree

The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[5, 4], [4, 3], [3, 2], [2, 1], [1, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.

View answer and explanation[5, 6]

Count normalized incident edges and select degree exactly one, obtaining [5, 6]. A singleton vertex has degree zero and is not a leaf under this convention.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0075/)

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





## 76. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 2, 3, 1]

The genuinely new edge closes the unique original path, giving [1, 2, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0076/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 77. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2] and E=[[1, 0], [0, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 2, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 2, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0077/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 78. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 2, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 2, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0078/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 79. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 3, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0079/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 80. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[2, 0, 3, 2]

The genuinely new edge closes the unique original path, giving [2, 0, 3, 2]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0080/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 81. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,0 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[0, 1, 2, 0]

The genuinely new edge closes the unique original path, giving [0, 1, 2, 0]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0081/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 82. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 3, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0082/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 83. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[2, 1, 0, 3, 2]

The genuinely new edge closes the unique original path, giving [2, 1, 0, 3, 2]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0083/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 84. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 2, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 2, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0084/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 85. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 3, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0085/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 86. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 4, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 4, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0086/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 87. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[2, 0, 3, 2]

The genuinely new edge closes the unique original path, giving [2, 0, 3, 2]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0087/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 88. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[2, 0, 4, 2]

The genuinely new edge closes the unique original path, giving [2, 0, 4, 2]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0088/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 89. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 4,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[3, 0, 4, 3]

The genuinely new edge closes the unique original path, giving [3, 0, 4, 3]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0089/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 90. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 0,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[0, 1, 2, 0]

The genuinely new edge closes the unique original path, giving [0, 1, 2, 0]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0090/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 91. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 3, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0091/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 92. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 4, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 4, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0092/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 93. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[2, 1, 0, 3, 2]

The genuinely new edge closes the unique original path, giving [2, 1, 0, 3, 2]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0093/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 94. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[2, 1, 0, 4, 2]

The genuinely new edge closes the unique original path, giving [2, 1, 0, 4, 2]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0094/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 95. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 4,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[3, 0, 4, 3]

The genuinely new edge closes the unique original path, giving [3, 0, 4, 3]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0095/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 96. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 0,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[0, 1, 2, 0]

The genuinely new edge closes the unique original path, giving [0, 1, 2, 0]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0096/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 97. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,0 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[0, 1, 2, 3, 0]

The genuinely new edge closes the unique original path, giving [0, 1, 2, 3, 0]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0097/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 98. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 2, 3, 1]

The genuinely new edge closes the unique original path, giving [1, 2, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0098/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 99. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 4,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[1, 0, 4, 1]

The genuinely new edge closes the unique original path, giving [1, 0, 4, 1]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0099/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate





## 100. Unique Cycle from a Genuinely New Tree Edge

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.

View answer and explanation[2, 1, 0, 4, 2]

The genuinely new edge closes the unique original path, giving [2, 1, 0, 4, 2]. A repeated existing edge is not an eligible new-edge cycle operation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261009-r02-tree-0100/)

Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261009-r02-tree-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261007-advance-20261009-r02-tree-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261009-r02-tree-foundations/validation-report.json)