Abstract
We give a new proof of Cayley's formula, which states that the number of labeled trees on n nodes is n n−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.
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