Central Limit Theorems for Additive Tree Parameters with Small Toll Functions

Abstract

We call a tree parameter additive if it can be determined recursively as the sum of the parameter values of all branches, plus a certain toll function. In this paper, we prove central limit theorems for very general toll functions, provided that they are bounded and small on average. Simply generated families of trees are considered as well as Pólya trees, recursive trees and binary search trees, and the results are illustrated by several examples of parameters for which we prove normal or log-normal limit laws.