Revisiting Shao and Sokal's $B_2$ index of phylogenetic balance

Abstract

Measures of phylogenetic balance, such as the Colless and Sackin indices, play an important role in phylogenetics. Unfortunately, these indices are specifically designed for phylogenetic trees, and do not extend naturally to phylogenetic networks (which are increasingly used to describe reticulate evolution). This led us to consider a lesser-known balance index, whose definition is based on a probabilistic interpretation that is equally applicable to trees and to networks. This index, known as the B 2 index, was first proposed by Shao and Sokal in 1990. Surprisingly, it does not seem to have been studied mathematically since. Likewise, it is used only sporadically in the biological literature, where it tends to be viewed as arcane and not very useful in practice -even though the evidence for this is scarce.In this paper, we study mathematical properties of B 2 such as its distribution under the most common models of random trees and its range over various classes of phylogenetic networks. We also assess its relevance in biological applications, and find it to be comparable to that of the Colless and Sackin indices. Altogether, our results call for a reevaluation of the status of this somewhat forgotten measure of phylogenetic balance.

Comparing the topology of phylogenetic network generators

“…The balance of phylogenetic trees is a well studied topological characteristic. The balance for phylogenetic networks is recently revisited in [3], where the B 2 balance for phylogenetic networks [46] is reintroduced. The B 2 balance is an index based on the entropy of probability distribution on leaf nodes.…”

“…For phylogenetic trees, it is well discussed which stochastic process generates phylogenetic trees that resemble trees reconstructed from data [6], and random tree generators are well compared from different points of view [48,40,12,3]. However, as increasingly many network generators are developed, there is a lack of discussion about the differences and similarities of generated random networks.…”

Tree balance indices: a comprehensive survey

“…While Shao and Sokal [73] originally defined the B 2 index for trees, Bienvenu et al [7] found that, due to its probabilistic interpretation, it is applicable to networks as well. In its definition, the value p x is the probability of reaching leaf x when starting at the root and assuming equiprobable branching at each inner vertex.…”

“…More precisely, the B 2 index is the Shannon entropy of the probability distribution (p x ) x∈VL(T ) . Bienvenu et al [7] described this vividly as letting water drip down from the root along the edges and then measure how evenly it is distributed among the leaves. Shao and Sokal [73] did not state what logarithm base to use (possibly log 10 considering their example), but Bienvenu et al [7] pointed out that base 2 is convenient when working with binary trees.…”

“…Bienvenu et al [7] described this vividly as letting water drip down from the root along the edges and then measure how evenly it is distributed among the leaves. Shao and Sokal [73] did not state what logarithm base to use (possibly log 10 considering their example), but Bienvenu et al [7] pointed out that base 2 is convenient when working with binary trees. It can be calculated using the function B2I from our R package treebalance.…”