CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-01

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.

Every tree with at least two vertices has at least two leaves

Statement

Every tree with at least two vertices has at least two distinct leaves.

Facts & Assumptions

Given: A tree $T$ with at least two vertices.

[L1]

Every two vertices of a tree are joined by a unique path (Equivalent characterisations of a nonempty tree by unique paths, edge count, minimal connectivity and maximal acyclicity).

[F1]

A leaf has degree one (Trees, forests, leaves and isolated vertices, Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

[F2]

Because $T$ is finite, its set of paths has a maximum-length member (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

Proof

technique · direct

1.1

Choose a path $P = v_{0} v_{1} \dots v_{k}$ of maximum length. Since $T$ has at least two vertices and is connected, $k \geq 1$ .

F2L1

2.1

If $v_{0}$ had a neighbour other than $v_{1}$ , that neighbour could not already lie on $P$ , since that would form a cycle; adjoining it would therefore lengthen the maximum-length path $P$ . Thus $v_{0}$ has degree one.

step 1.1L1

2.2

The same argument shows that $v_{k}$ has degree one.

step 1.1L1

3.1

Since $v_{0} \neq v_{k}$ , these are two distinct leaves.

step 1.1step 2.1step 2.2F1∎

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)
ISI Bangalore discrete mathematics notes, Trees and Cayley’s theorem (standard reference, not scraped)