Hereditary tree growth and Levy forests

TL;DR

This paper develops a framework for hereditary properties of rooted real trees, introduces growth processes of Galton-Watson trees, and proves their convergence to Levy forests, providing new insights into tree structures and their limits.

Contribution

It introduces hereditary reductions for real trees, characterizes growth processes of Galton-Watson trees, and establishes their convergence to Levy forests, linking discrete and continuous models.

Findings

Hereditary properties lead to a tightness criterion in the space of trees.

Growth processes of Galton-Watson trees converge to Levy forests.

Characterization of Levy forests via leaf-length erasure.

Abstract

We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction. We only consider the metric structure of trees, and our framework is the space $\bT$ of pointed isometry classes of locally compact rooted real trees equipped with the Gromov-Hausdorff distance. Some of the main results of the paper are a general tightness criterion in $\bT$ and limit theorems for growing families of trees. We apply these results to Galton-Watson trees with exponentially distributed edge lengths. This class is preserved by hereditary reduction. Then we consider families of such Galton-Watson trees that are consistent under hereditary reduction and that we call growth processes. We prove that the associated families of offspring distributions are completely characterised by the branching mechanism of a continuous-state branching process. We also prove that such growth processes converge to Levy forests. As a by-product of this convergence, we obtain a characterisation of the laws of Levy forests in terms of leaf-length erasure and we obtain invariance principles for discrete Galton-Watson trees, including the super-critical cases.