Tree limits and limits of random trees

TL;DR

This paper investigates the limits of various classes of random trees, providing general theorems for their convergence to tree limits, including models like Galton-Watson, split trees, and trees from branching processes.

Contribution

It introduces broad theorems for the convergence of multiple classes of random trees to tree limits, expanding understanding of their asymptotic behavior.

Findings

Tree limits exist for many classes of random trees.

General theorems cover Galton-Watson, split, and branching process trees.

Results include convergence for labeled, ordered, recursive, and preferential attachment trees.

Abstract

We explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton-Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.