# Cayley's formula

The complete list of all trees on 2,3,4 labeled vertices: $2^{2-2}=1$ tree with 2 vertices, $3^{3-2}=3$ trees with 3 vertices and $4^{4-2}=16$ trees with 4 vertices.

In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for any positive integer n, the number of trees on n labeled vertices is $n^{n-2}$.

The formula equivalently counts the number of spanning trees of a complete graph with labeled vertices (sequence A000272 in OEIS).

## Proof

Many remarkable proofs of Cayley's tree formula are known.[1] One classical proof of the formula uses Kirchhoff's matrix tree theorem. Prüfer sequences yield a bijective proof of Cayley's formula. Another bijective proof, by André Joyal, finds a one-to-one transformation between n-node trees with two distinguished nodes and maximal directed pseudoforests. A proof by double counting due to Jim Pitman applies to trees.

## History

The formula was first discovered by Carl Wilhelm Borchardt in 1860, and proved via a determinant.[2] In a short 1889 note, Cayley extended the formula in several directions, by taking into account the degrees of the vertices.[3] Although he referred to Borchardt's original paper, the name "Cayley's formula" became standard in the field.