Cayley's formula

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The complete list of all trees on 2,3,4 labeled vertices: tree with 2 vertices, trees with 3 vertices and 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 .

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.

References

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