Cophenetic metrics for phylogenetic trees, after Sokal and Rohlf

TL;DR

This paper introduces a new family of cophenetic metrics for weighted phylogenetic trees, based on encoding trees with cophenetic values and comparing these vectors using $L^p$ norms, to improve tree comparison methods.

Contribution

It formally defines and studies a novel family of cophenetic metrics for weighted phylogenetic trees, extending previous ideas to nested taxa and arc weights.

Findings

Metrics have well-defined properties like neighbors and diameter.

Distribution and rank correlation of metrics analyzed.

Numerical and analytical studies demonstrate their effectiveness.

Abstract

Phylogenetic tree comparison metrics are an important tool in the study of evolution, and hence the definition of such metrics is an interesting problem in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed to measure quantitatively the difference between a pair of phylogenetic trees by first encoding them by means of their half-matrices of cophenetic values, and then comparing these matrices. This idea has been used several times since then to define dissimilarity measures between phylogenetic trees but, to our knowledge, no proper metric on weighted phylogenetic trees with nested taxa based on this idea has been formally defined and studied yet. Actually, the cophenetic values of pairs of different taxa alone are not enough to single out phylogenetic trees with weighted arcs or nested taxa. In this paper we define a family of cophenetic metrics that compare phylogenetic trees on a same set of taxa by encoding them by means of their vectors of cophenetic values of pairs of taxa and depths of single taxa, and then computing the $L^p$ norm of the difference of the corresponding vectors. Then, we study, either analytically or numerically, some of their basic properties: neighbors, diameter, distribution, and their rank correlation with each other and with other metrics.