Cayley's formula

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

Not to be confused with Cayley's theorem.

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

Proof[edit]

Many proofs of Cayley's tree formula are known.^[1] One classical proof of the formula uses Kirchhoff's matrix tree theorem, a formula for the number of spanning trees in an arbitrary graph involving the determinant of a matrix. 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 counts in two different ways the number of different sequences of directed edges that can be added to an empty graph on n vertices to form from it a rooted tree; see Double counting (proof technique) § Counting trees.

History[edit]

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.