A generalized enumeration of labeled trees and reverse Prüfer algorithm

A leader of a tree T   on [n][n] is a vertex which has no smaller descendants in T. Gessel and Seo showed that∑T∈Tnu(#ofleadersinT)c(degreeof1inT)=uPn−1(1,u,cu), which is a generalization of Cayley's formula, where TnTn is the set of trees on [n][n] andPn(a,b,c)=c∏i=1n−1(ia+(n−i)b+c). Using a variation of the Prüfer code which is called a RP-code, we give a simple bijective proof of Gessel and Seo's formula.