A new proof of Cayley's formula for counting labeled trees

Abstract

We give a new proof of Cayley's formula, which states that the number of labeled trees on n nodes is nn-2. This proof uses a difficult combinatorial identity, and it could equally well be regarded as a proof of this identity that uses Cayley's formula. The proof proceeds by counting labeled rooted trees with n vertices and j improper edges, where an improper edge is one whose endpoint closer to the root has a larger label than some vertex in the subtree rooted on the edge. © 1995.