A central limit theorem for additive functionals of increasing trees

Abstract

AbstractA tree functional is called additive if it satisfies a recursion of the form$F(T) = \sum_{j=1}^k F(B_j) + f(T)$, whereB1, …,Bkare the branches of the treeTandf(T) is a toll function. We prove a general central limit theorem for additive functionals ofd-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalized plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group ofd-ary increasing trees, but other examples (old and new) are covered as well.