False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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: $\tau(K_n)=n^{n-2}$ for $n\ge2$ , with $\tau(K_1)=1$ and $\tau(K_0)=0$ ).

[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 $16\ne2$ , Cayley's formula does not count unlabelled isomorphism types.

step 1.1step 1.2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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