A note on limits of sequences of binary trees

TL;DR

This paper introduces a new notion of convergence for binary trees based on subtree sizes, explores the structure of the limit space, and establishes a central limit theorem for random trees under this topology.

Contribution

It develops a subtree size-based convergence framework for binary trees, characterizes the limit space, and proves a central limit theorem for sequences of random trees.

Findings

Characterization of the set of possible limits of binary trees

Structural description of the limit space as a metric space

Central limit theorem with mixed asymptotic normality for random trees

Abstract

We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a characterization of the set of possible limits and its structure as a metric space. For random trees the subtree size topology arises in the context of algorithms for searching and sorting when applied to random input, resulting in a sequence of nested trees. For these we obtain a structural result based on a local version of exchangeability. This in turn leads to a central limit theorem, with possibly mixed asymptotic normality.