# 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 .

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

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.

## Other properties

Cayley's formula immediately gives the number of labelled rooted forests on *n* vertices, namely (*n*+1)^{n-1}.
Each labelled rooted forest can be turned into a labelled tree with one extra vertex, by adding a vertex with label *n*+1 and connecting
it to all roots of the trees in the forest.

There is a close connection with rooted forests and parking functions, since the number of parking functions on *n*
cars is also (*n*+1)^{n-1}. A bijection between rooted forests and parking functions was given by M. P. Schützenberger in 1968.

## References

- ↑ Aigner, Martin; Ziegler, Günter M. (1998).
*Proofs from THE BOOK*. Springer-Verlag. pp. 141–146. - ↑ Borchardt, C. W. (1860). "Über eine Interpolationsformel für eine Art Symmetrischer Functionen und über Deren Anwendung".
*Math. Abh. der Akademie der Wissenschaften zu Berlin*: 1–20. - ↑ Cayley, A. (1889). "A theorem on trees".
*Quart. J. Math*.**23**: 376–378.