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A note on limits of sequences of binary trees

Rudolf Grübel

TL;DR

A notion of convergence for binary trees that is based on subtree sizes is discussed, which leads to a central limit theorem, with possibly mixed asymptotic normality.

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.

A note on limits of sequences of binary trees

TL;DR

A notion of convergence for binary trees that is based on subtree sizes is discussed, which leads to a central limit theorem, with possibly mixed asymptotic normality.

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.
Paper Structure (4 sections, 6 theorems, 42 equations)

This paper contains 4 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Subtree size convergence
  3. Local exchangeability
  4. A second order result

Key Result

Lemma 1

Let $\psi:\mathbb{\space V\space}\to [0,1]$ be such that $\psi(\emptyset)=1$ and Then there exists a unique $\mu\in\mathfrak{M}_\infty$ such that $\mu(B_u)=\psi(u)$ for all $u\in\mathbb{\space V\space}$.

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Example 3
  • Example 4
  • Example 5
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 4 more