Squaring within the Colless index yields a better balance index

TL;DR

This paper introduces a modified phylogenetic balance index that replaces absolute differences with squared differences, improving its discriminatory power and analytical tractability over the original Colless index.

Contribution

Replacing absolute differences with squared differences in the Colless index creates a more powerful and analytically manageable phylogenetic balance index.

Findings

The squared difference index better distinguishes balanced trees.

It overcomes the original index's limitations in identifying maximally balanced trees.

The new index is more discriminating and easier to analyze statistically.

Abstract

The Colless index for bifurcating phylogenetic trees, introduced by Colless (1982), is defined as the sum, over all internal nodes $v$ of the tree, of the absolute value of the difference of the sizes of the clades defined by the children of $v$. It is one of the most popular phylogenetic balance indices, because, in addition to measuring the balance of a tree in a very simple and intuitive way, it turns out to be one of the most powerful and discriminating phylogenetic shape indices. But it has some drawbacks. On the one hand, although its minimum value is reached at the so-called maximally balanced trees, it is almost always reached also at trees that are not maximally balanced. On the other hand, its definition as a sum of absolute values of differences makes it difficult to study analytically its distribution under probabilistic models of bifurcating phylogenetic trees. In this paper we show that if we replace in its definition the absolute values of the differences of clade sizes by the squares of these differences, all these drawbacks are overcome and the resulting index is still more powerful and discriminating than the original Colless index.