TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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Cayley's formula: $\tau(K_n)=n^{n-2}$ for $n\ge2$ , with $\tau(K_1)=1$ and $\tau(K_0)=0$

Statement

For $n\ge2$ , the complete graph $K_n$ has

$\tau(K_n)=n^{n-2}$

spanning trees. Moreover, $\tau(K_1)=1$ and $\tau(K_0)=0$ .

Facts & Assumptions

Given: A natural number $n$ .

[L1]

Trees on the fixed label set $n$ are in bijection with words of length $n-2$ on $n$ when $n\ge2$ (Prüfer encoding and decoding are inverse bijections between labelled trees on $n\ge2$ vertices and words of length $n-2$ on their labels).

[L2]

The set of functions from an $m$ -element set to an $n$ -element set has cardinality $n^m$ (The set $A^{B}$ of functions $B \to A$ between finite sets is finite, with $\lvert A^{B}\rvert = \lvert A\rvert^{\lvert B\rvert}$ , Exponentiation of natural numbers, $m^{n}$ , and its agreement with the integer power in $\mathbb{R}$ ).

[F1]

Every labelled tree on $n$ is a spanning subgraph of $K_n$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices, The spanning-tree number $\tau(G)$ ).

Proof

technique · direct

1.1

For $n\ge2$ , spanning trees of $K_n$ are exactly the trees on its fixed label set.

2.1

By the Prüfer bijection, they correspond to functions from an $(n-2)$ -element position set to $n$ , of which there are $n^{n-2}$ .

step 1.1L1L2

3.1

The sole spanning tree of $K_1$ is $K_1$ itself, so $\tau(K_1)=1$ . The null graph $K_0$ is not a tree and has no spanning tree, so $\tau(K_0)=0$ .

F1∎

Depends on

Used by

τ(K₄)=16 by direct Prüfer enumeration and Cayley's formula Example
Cayley's formula counts unlabelled trees False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 78 results over 26 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)

Cayley's formula: τ(Kn)=nn−2\tau(K_n)=n^{n-2}τ(Kn​)=nn−2 for n≥2n\ge2n≥2, with τ(K1)=1\tau(K_1)=1τ(K1​)=1 and τ(K0)=0\tau(K_0)=0τ(K0​)=0