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.
Shor, P. W. (1995). A new proof of Cayley’s formula for counting labeled trees. Journal of Combinatorial Theory, Series A, 71(1), 154–158. https://doi.org/10.1016/0097-3165(95)90022-5