Alphabeta MathTrees, Forests and Spanning Trees — Examples
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How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 12 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 12 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Combinatorics
Real Analysis
2 · Logical flowchart
3 · Items
Example
The isomorphism types of trees on at most five vertices
Example
A labelled tree and its Prüfer code computed in both directions
Example
τ(Cₙ)=n for every n≥3
Example
τ(K₄)=16 by direct Prüfer enumeration and Cayley's formula
Example
A fundamental cycle and a fundamental cut in a fixed spanning tree
Example
A connected graph with two distinct spanning trees
Example
Kruskal's and Prim's procedures on the same weighted graph
Counterexample
An acyclic graph need not be a tree
Counterexample
A graph on n vertices with n-1 edges need not be a tree
Counterexample
A weighted graph with two distinct minimum spanning trees
False statement
Cayley's formula counts unlabelled trees
False statement