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Maximum Agreement Subtrees and Hölder homeomorphisms between Brownian trees

Thomas Budzinski, Delphin Sénizergues

Abstract

We prove that the size of the largest common subtree between two uniform, independent, leaf-labelled random binary trees of size $n$ is typically less than $n^{1/2-\varepsilon}$ for some $\varepsilon>0$. Our proof relies on the coupling between discrete random trees and the Brownian tree and on a recursive decomposition of the Brownian tree due to Aldous. Along the way, we also show that almost surely, there is no $(1-\varepsilon)$-Hölder homeomorphism between two independent copies of the Brownian tree.

Maximum Agreement Subtrees and Hölder homeomorphisms between Brownian trees

Abstract

We prove that the size of the largest common subtree between two uniform, independent, leaf-labelled random binary trees of size is typically less than for some . Our proof relies on the coupling between discrete random trees and the Brownian tree and on a recursive decomposition of the Brownian tree due to Aldous. Along the way, we also show that almost surely, there is no -Hölder homeomorphism between two independent copies of the Brownian tree.
Paper Structure (41 sections, 19 theorems, 109 equations, 5 figures)

This paper contains 41 sections, 19 theorems, 109 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Maximum agreement subtree.
  3. Maximum agreement subtree of random trees.
  4. Comparison with the Brownian tree.
  5. Hölder homeomorphisms of Brownian tree.
  6. Recursive decomposition of the Brownian tree.
  7. Ideas of the proof of Proposition \ref{['prop_homeo_mass']}.
  8. Structure of the paper.
  9. Acknowledgments.
  10. Definitions and preliminaries
  11. Labelled binary trees and maximum agreement subtree
  12. The Brownian tree
  13. Dirichlet and Beta distributions
  14. Coupling between discrete and continuum random trees
  15. The Aldous recursive decomposition
  16. ...and 26 more sections

Key Result

Theorem 1

For all $n \geq 3$, let $T_n$ and $T'_n$ be two independent uniform labelled binary trees of size $n$. There exists a constant $\varepsilon_{thm_MAST_upper}>0$ such that we have

Figures (5)

  • Figure 1: Two labelled binary trees $t$ an $t'$ and their largest common subtree, induced by the set $I=\{1,2,4,5,8\}$.
  • Figure 2: Coupling between the $5$-pointed Brownian tree $(\widetilde{\mathcal{T}},X_1^1,\dots, X_5^1)$ and $T_5$. The combinatorial structure of the paths joining the distinguished points, shown in blue, is that of the discrete tree $T_5$ we started with.
  • Figure 3: A decomposed Brownian tree $\mathcal{T}$ and its decomposition $(\mathcal{R}[\mathbf{i}])_{\mathbf{i} \in \mathbb{T}_3^2}$ of depth $2$.
  • Figure 4: One step of the decomposition for a region $\mathcal{R}[\mathbf{i}]$ delimited by respectively one point and two points. Note that in both cases, the newly created region $\mathcal{R}[\mathbf{i} 3]$ is delimited by only one point.
  • Figure 5: The sequence $(b_h)_{1\leq h \leq \ell}$ and their positions relative to $\mathcal{R}[\mathbf{i}_{j_h}2]$ and $\mathcal{R}[\mathbf{i}_{j_h}3]$.

Theorems & Definitions (42)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • proof
  • ...and 32 more