  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. In this
paper, we study mathematical properties of $B_2$ such as its expectation and
variance under the most common models of random trees and its extremal values
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.
