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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
No
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Nonempty finite trees: connectivity, cycles, and unique paths · qf-tree-added-edge-cycle · candidate
Canonical question JSONL · English JSONL · Validation report