Asymptotic analysis of Hoppe trees

TL;DR

This paper studies a new random tree model inspired by Hoppe's urn, analyzing its height, path length, leaves, and node depth as the tree grows large, providing expectations, variances, and asymptotic distributions.

Contribution

It introduces the Hoppe tree model and derives its asymptotic properties, extending known results from recursive trees to this more general setting.

Findings

Asymptotic distributions for height and path length

Explicit formulas for expectations and variances

Behavior of the last inserted node's depth

Abstract

We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight $\vartheta>0$, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For $\vartheta=1$ the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length and number of leaves of the Hoppe tree with $n$ nodes as well as the depth of the last inserted node asymptotically as $n\to \infty$. Mainly expectations, variances and asymptotic distributions of these parameters are derived.