The largest common subtree of two random trees

TL;DR

This work analyzes the largest common subtree (LCS) of two independent Bienaymé trees conditioned to have size $n$, revealing a $\sqrt{n}$-scaling in the critical regime with moment conditions and characterizing the limit via a Phi functional applied to two independent Brownian CRTs. The authors develop a multifaceted approach: plane-tree encodings, a bootstrap to bound rooted LCS, a spine-based deterministic-probabilistic decomposition, and a skeleton reduction to a bounded-leaf problem, followed by a scaling limit to CRTs. They prove that the limit equals a constant times the LCS of two CRTs, and provide a tightness argument showing necessity of the moment condition on at least one offspring distribution. The methodology combines deep probabilistic tools (local limit theorems, big-jump principles, Many-to-One) with fine geometric analysis on real trees, delivering a comprehensive framework for LCS in conditioned random trees and opening several directions for future exploration. The results have implications for understanding structural overlaps in random trees and may inform related problems in phylogenetic combinatorics and random graph topology, where substructure coincidence under scaling limits is central.

Abstract

We study the size and structure of the largest common subtree (LCS) between two independent Bienaymé trees conditioned to have size $n$. When the trees are critical with finite $2$nd and $(2+κ)$th moment respectively for some $κ>0$, we prove that the LCS has size of order $\sqrt{n}$, and is approximated by the length of three paths meeting at a central node. Moreover, we show that the largest common subtree between two critical independent Bienaymé trees with size $n$ and finite second moments may be much larger than $\sqrt{n}$, implying that our result is tight. We also pose a number of open questions and suggestions for future research.

Figures (6)

• Figure 1: Two real trees with their largest common Y-shaped subtree highlighted red.
• Figure 2: Illustration of the spinal decomposition of $\mathtt{T}^\bullet$.
• Figure 3: Illustration of the four bad cases: (from left to right) sausages/twigs, skewers, flowers and bushes. Gray trees are larger than micro-sized.
• Figure 4: On the left, a depiction of the common subtree $\mathtt{T}$. The black vertices are the vertices in $\mathtt{R}$, and $\mathrm{Collap}(\mathtt{R})$ is depicted on the right. The red triangles represent the trees in $\mathtt{T}$ off vertices in $\mathrm{Collap}(\mathtt{R})$ and the blue triangles represent the trees in $\mathtt{T}$ off the vertices in $\mathtt{R}\setminus\mathrm{Collap}(\mathtt{R})$.
• Figure 5: We partition the path from $u$ to $v$ in $\tau$ and the path from $u'$ to $v'$ in $\tau'$, of the same length, into $7$ parts.

Theorems & Definitions (84)

• Theorem 1.1
• Theorem 1.2
• Remark
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.1
• Proposition 2.2: Many-to-One principle