A new proof of Cayleyʹs formula for counting labeled trees

We give a new proof of Cayleyʹs 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 Cayleyʹs 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.