Asymptotics of heights in random trees constructed by aggregation

TL;DR

This paper investigates the asymptotic behavior of the height and subtrees of a class of growing random trees constructed by sequentially attaching segments, focusing on cases where the segment lengths follow a regularly varying sequence.

Contribution

It provides the first detailed asymptotic analysis of the height and subtree structures for trees built with regularly varying segment lengths, extending previous work on compactness and dimension.

Findings

Asymptotic formulas for the height of the trees.

Distributional limits for subtrees spanned by random points.

Characterization of growth behavior based on the regular variation index.

Abstract

To each sequence $(a_n)$ of positive real numbers we associate a growing sequence $(T_n)$ of continuous trees built recursively by gluing at step $n$ a segment of length $a_n$ on a uniform point of the pre-existing tree, starting from a segment $T_1$ of length $a_1$. Previous works on that model focus on the influence of $(a_n)$ on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence $(a_n)$ is regularly varying with a non-negative index, so that the sequence $(T_n)$ exploses. We determine the asymptotics of the height of $T_n$ and of the subtrees of $T_n$ spanned by the root and $\ell$ points picked uniformly at random and independently in $T_n$, for all $\ell \in \mathbb N$.