Measuring tree balance using symmetry nodes -- a new balance index and its extremal properties

TL;DR

This paper introduces the symmetry nodes index, a new measure for tree balance based on symmetry nodes, analyzes its properties, compares it with existing indices, and provides an efficient R implementation.

Contribution

The paper presents the first symmetry nodes based balance index, analyzes its extremal properties, and offers an R package for practical computation.

Findings

The symmetry nodes index generalizes the cherry index and specializes Rogers' J index.

It can be computed in linear time.

The index has well-defined extremal properties.

Abstract

Effects like selection in evolution as well as fertility inheritance in the development of populations can lead to a higher degree of asymmetry in evolutionary trees than expected under a null hypothesis. To identify and quantify such influences, various balance indices were proposed in the phylogenetic literature and have been in use for decades. However, so far no balance index was based on the number of \emph{symmetry nodes}, even though symmetry nodes play an important role in other areas of mathematical phylogenetics and despite the fact that symmetry nodes are a quite natural way to measure balance or symmetry of a given tree. The aim of this manuscript is thus twofold: First, we will introduce the \emph{symmetry nodes index} as an index for measuring balance of phylogenetic trees and analyze its extremal properties. We also show that this index can be calculated in linear time. This new index turns out to be a generalization of a simple and well-known balance index, namely the \emph{cherry index}, as well as a specialization of another, less established, balance index, namely \emph{Rogers' $J$ index}. Thus, it is the second objective of the present manuscript to compare the new symmetry nodes index to these two indices and to underline its advantages. In order to do so, we will derive some extremal properties of the cherry index and Rogers' $J$ index along the way and thus complement existing studies on these indices. Moreover, we used the programming language \textsf{R} to implement all three indices in the software package \textsf{symmeTree}, which has been made publicly available.