A generalized enumeration of labeled trees and reverse Prüfer algorithm

A leader of a tree T on [ n ] is a vertex which has no smaller descendants in T. Gessel and Seo showed that ∑ T ∈ T n u ( # of leaders in T ) c ( degree of 1 in T ) = u P n − 1 ( 1 , u , c u ) , which is a generalization of Cayleyʹs formula, where T n is the set of trees on [ n ] and P n ( a , b , c ) = c ∏ i = 1 n − 1 ( i a + ( 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 Seoʹs formula.