Schr\"oder's problems and scaling limits of random trees

TL;DR

This paper investigates the average structure of certain combinatorial bracketings, showing that as their size grows, they converge to the Brownian continuum random tree, revealing universal scaling limits.

Contribution

It proves that the uniform random bracketings in Schr"oder's four problems converge to the Brownian CRT as size increases, establishing a universal scaling limit.

Findings

Uniform random bracketings converge to the Brownian CRT

Scaling limits are universal across the four problems

As size increases, the structure approaches a continuous random tree

Abstract

In a classic paper Schr\"oder posed four combinatorial problems about the number of certain types of bracketings of words and sets. Here we address what these bracketings look like on average. For each of the four problems we prove that a uniform pick from the appropriate set of bracketings, when considered as a tree, has the Brownian continuum random tree as its scaling limit as the size of the word or set goes to infinity.