Cayley's formula

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.

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.

Cayley's formula immediately gives the number of labelled rooted forestsonn 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.^[4]

The following generalizes Cayley's formula to labelled forests: Let T_n,k be the number of labelled forests on n vertices with k connected components, such that vertices 1, 2, ..., k all belong to different connected components. Then T_n,k = k n^{n − k − 1}.^[5]