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.

A tree on $n \geq 1$ vertices has $n - 1$ edges

Statement

A tree with $n \geq 1$ vertices has exactly $n - 1$ edges.

Facts & Assumptions

Given: A tree $T$ with $∣ V (T) ∣ = n$ .

[L1]

Every forest satisfies $∣ V ∣ = ∣ E ∣ + c$ (For every forest, $∣ V ∣ = ∣ E ∣ + c$ , where $c$ is the number of connected components).

[F1]

A tree is connected and acyclic, so it is a forest with exactly one connected component (Trees, forests, leaves and isolated vertices, Connected graphs and connected components defined by the existence of vertex paths).

Proof

technique · direct

1.1

Apply the forest identity to $T$ : $n = ∣ E (T) ∣ + 1$ .

L1F1

2.1

Therefore $∣ E (T) ∣ = n - 1$ .

step 1.1∎

Depends on

Used by

Dependency tree · two levels

19 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)

A tree on n≥1n≥1 vertices has n−1n−1 edges