The CRT is the scaling limit of unordered binary trees

TL;DR

This paper proves that as the size of a uniform unordered binary tree grows, its shape converges to the Brownian continuum random tree, linking discrete tree models to a continuous limit in a rigorous way.

Contribution

It establishes the scaling limit of uniform unordered binary trees as the Brownian CRT, extending known results from plane and labeled trees.

Findings

Uniform unordered binary trees converge to the Brownian CRT in the Gromov-Hausdorff topology.

The analysis uses combinatorial and probabilistic methods involving tree trimming procedures.

The result aligns the behavior of unordered binary trees with other tree models in the continuum limit.

Abstract

We prove that a uniform, rooted unordered binary tree with $n$ vertices has the Brownian continuum random tree as its scaling limit for the Gromov-Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform plane trees or labeled trees. Our analysis rests on a combinatorial and probabilistic study of appropriate trimming procedures of trees.