Nodal distances for rooted phylogenetic trees

TL;DR

This paper introduces a new method for measuring dissimilarity between rooted phylogenetic trees by splitting path lengths, enabling metrics to distinguish all such trees with weighted arcs, extending previous limitations.

Contribution

It proposes a novel splitting technique for path lengths that uniquely identifies any rooted phylogenetic tree with weighted arcs, allowing for comprehensive metric definitions.

Findings

The new metrics distinguish all rooted phylogenetic trees with positive real arc weights.

Splitting path lengths into two parts resolves previous non-distinguishability issues.

Basic properties of these metrics are established using $L^p$ norms.

Abstract

Dissimilarity measures for (possibly weighted) phylogenetic trees based on the comparison of their vectors of path lengths between pairs of taxa, have been present in the systematics literature since the early seventies. But, as far as rooted phylogenetic trees goes, these vectors can only separate non-weighted binary trees, and therefore these dissimilarity measures are metrics only on this class. In this paper we overcome this problem, by splitting in a suitable way each path length between two taxa into two lengths. We prove that the resulting splitted path lengths matrices single out arbitrary rooted phylogenetic trees with nested taxa and arcs weighted in the set of positive real numbers. This allows the definition of metrics on this general class by comparing these matrices by means of metrics in spaces of real-valued $n\times n$ matrices. We conclude this paper by establishing some basic facts about the metrics for non-weighted phylogenetic trees defined in this way using $L^p$ metrics on these spaces of matrices.