Trees with power-like height dependent weight

TL;DR

This paper analyzes a class of planar rooted random trees with height-dependent weights, deriving their asymptotic properties and showing their local limit converges to the Uniform Infinite Planar Tree regardless of the height distribution exponent.

Contribution

It provides a detailed analysis of the generating function's analyticity and establishes the asymptotic form and local limit of these weighted trees.

Findings

Asymptotic form of the total weight $Z_N$ for large trees

Local limit at large size is the Uniform Infinite Planar Tree

Independence of the local limit from the height distribution exponent $eta$

Abstract

We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is given by a power function $h^\alpha$. Defining the total weight for such trees of fixed size to be $Z_N$, a detailed analysis of the analyticity properties of the corresponding generating function is provided. Based on this, we determine the asymptotic form of $Z_N$ and show that the local limit at large size is identical to the Uniform Infinite Planar Tree, independent of the exponent $\alpha$ of the height distribution function.