A new balance index for phylogenetic trees

TL;DR

This paper introduces the total cophenetic index, a new measure for phylogenetic tree balance that is efficient to compute, more discriminative than existing indices, and provides formulas for its extremal and expected values under common evolutionary models.

Contribution

The paper defines and analyzes the total cophenetic index, offering new formulas for its extremal and expected values, and compares its effectiveness to existing indices.

Findings

The total cophenetic index can be computed in linear time.

It has a larger range and greater resolution than Colless' and Sackin's indices.

Exact formulas for its maximum, minimum, and expected values under Yule and uniform models are provided.

Abstract

Several indices that measure the degree of balance of a rooted phylogenetic tree have been proposed so far in the literature. In this work we define and study a new index of this kind, which we call the total cophenetic index: the sum, over all pairs of different leaves, of the depth of their least common ancestor. This index makes sense for arbitrary trees, can be computed in linear time and it has a larger range of values and a greater resolution power than other indices like Colless' or Sackin's. We compute its maximum and minimum values for arbitrary and binary trees, as well as exact formulas for its expected value for binary trees under the Yule and the uniform models of evolution. As a byproduct of this study, we obtain an exact formula for the expected value of the Sackin index under the uniform model, a result that seems to be new in the literature.