A Generalized Enumeration of Labeled Trees and Reverse Pr\"ufer Algorithm
Abstract
A leader of a tree T on [n] is a vertex which has no smaller descendants in T. Gessel and Seo showed ΣT ∈ Tnu(# of leaders in T) c(degree of 1 in T)=u Pn-1(1,u,cu), which is a generalization of Cayley formula, where Tn is the set of trees on [n] and Pn(a,b,c)=cΠi=1n-1(ia+(n-i)b+c). Using a variation of Pr\"ufer code which is called a RP-code, we give a simple bijective proof of Gessel and Seo's formula.
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