Random recursive trees: A boundary theory approach

TL;DR

This paper introduces a boundary theory approach to analyze the convergence and limit properties of random recursive trees, providing new proofs and representations for their asymptotic behavior.

Contribution

It presents a novel boundary theory framework for recursive trees, enabling direct proofs of convergence and new limit theorems for tree functionals.

Findings

Proves convergence of random recursive trees via Doob-Martin compactification

Provides a representation of the limit in terms of input sequences

Derives strong limit theorems for path length and Wiener index

Abstract

We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in terms of the input sequence of the algorithm. We further show that this approach can be used to obtain strong limit theorems for various tree functionals, such as path length or the Wiener index.