Additive functionals of $d$-ary increasing trees

TL;DR

This paper establishes a central limit theorem for additive functionals of d-ary increasing trees and applies it to various examples, including the size of automorphism groups, using a unified method.

Contribution

It introduces a general CLT for additive tree functionals and extends the approach to generalized plane-oriented increasing trees, covering new and existing applications.

Findings

Proves a log-normal distribution law for automorphism group sizes.

Demonstrates the CLT applies to a broad class of additive functionals.

Provides a unified framework for analyzing various tree functionals.

Abstract

A tree functional is called additive if it satisfies a recursion of the form $F(T) = \sum_{j=1}^k F(B_j) + f(T)$, where $B_1,\ldots,B_k$ are the branches of the tree $T$ and $f(T)$ is a toll function. We prove a general central limit theorem for additive functionals of $d$-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalised 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 of $d$-ary increasing trees, but many other examples (old and new) are covered as well.