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 every 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, 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

1. Aigner, Martin; Ziegler, Günter M. (1998). Proofs from THE BOOK. Springer-Verlag. pp. 141–146.
2. 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.
3. Cayley, A. (1889). "A theorem on trees". Quart. J. Math 23: 376–378.