False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-01

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Cayley's formula counts unlabelled trees

False Statement

Cayley's number $n^{n - 2}$ is the number of unlabelled trees on $n$ vertices.

Facts & Assumptions

Given: The claimed interpretation of Cayley's formula.

[L1]

Cayley's formula counts spanning trees of the labelled complete graph $K_{n}$ (Cayley's formula: $τ (K_{n}) = n^{n - 2}$ for $n \geq 2$ , with $τ (K_{1}) = 1$ and $τ (K_{0}) = 0$ ).

[L2]

Up to isomorphism, there are only two trees on four vertices: $P_{4}$ and $K_{1, 3}$ (The isomorphism types of trees on at most five vertices).

Refutation

technique · direct

1.1

At $n = 4$ , Cayley's number is $4^{4 - 2} = 16$ .

1.2

The unlabelled count is $2$ .

2.1

Since $16 \neq 2$ , Cayley's formula does not count unlabelled isomorphism types.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

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

ISI Bangalore discrete mathematics notes, Trees and Cayley’s theorem (standard reference, not scraped)