Recovering a tree from the lengths of subtrees spanned by a randomly chosen sequence of leaves

TL;DR

This paper investigates whether a tree's structure can be uniquely identified from the distribution of subtree lengths spanned by randomly chosen leaves, showing positive results for specific tree families.

Contribution

It demonstrates that under certain conditions and for specific tree families, the joint distribution of subtree lengths uniquely determines the tree structure.

Findings

Unique identification for trees with edges in general position

Identification for ultrametric trees

Identification for trees with equal edge weights in certain families

Abstract

Given an edge-weighted tree $T$ with $n$ leaves, sample the leaves uniformly at random without replacement and let $W_k$, $2 \le k \le n$, be the length of the subtree spanned by the first $k$ leaves. We consider the question, "Can $T$ be identified (up to isomorphism) by the joint probability distribution of the random vector $(W_2, \ldots, W_n)$?" We show that if $T$ is known {\em a priori} to belong to one of various families of edge-weighted trees, then the answer is, "Yes." These families include the edge-weighted trees with edge-weights in general position, the ultrametric edge-weighted trees, and certain families with equal weights on all edges such as $(k+1)$-valent and rooted $k$-ary trees for $k \ge 2$ and caterpillars.