`Lassoing' a phylogenetic tree I: Basic properties, shellings, and covers

TL;DR

This paper explores the minimal subsets of leaf-to-leaf genetic distances needed to uniquely reconstruct phylogenetic trees, addressing practical limitations in data collection and estimation.

Contribution

It introduces new theoretical insights into which distance subsets suffice for tree reconstruction, extending classical results to more realistic data scenarios.

Findings

Identifies conditions under which a subset of distances can reconstruct a tree

Provides methods for determining minimal lasso sets

Enhances understanding of data requirements in phylogenetic inference

Abstract

A classical result, fundamental to evolutionary biology, states that an edge-weighted tree $T$ with leaf set $X$, positive edge weights, and no vertices of degree 2 can be uniquely reconstructed from the set of leaf-to-leaf distances between any two elements of $X$. In biology, $X$ corresponds to a set of taxa (e.g. extant species), the tree $T$ describes their phylogenetic relationships, the edges correspond to earlier species evolving for a time until splitting in two or more species by some speciation/bifurcation event, and their length corresponds to the genetic change accumulating over that time in such a species. In this paper, we investigate which subsets of $\binom{X}{2}$ suffice to determine (`lasso') a tree from the leaf-to-leaf distances induced by that tree. The question is particularly topical since reliable estimates of genetic distance - even (if not in particular) by modern mass-sequencing methods - are, in general, available only for certain combinations of taxa.