Sound Colless-like balance indices for multifurcating trees

The Colless index is one of the most popular and natural balance indices for bifurcating phylogenetic trees, but it makes no sense for multifurcating trees. In this paper we propose a family of Colless-like balance indices $\mathfrak{C}_{D,f}$, which depend on a dissimilarity $D$ and a function $f:\mathbb{N}\to \mathbb{R}_{\geq 0}$, that generalize the Colless index to multifurcating phylogenetic trees. We provide two functions $f$ such that the most balanced phylogenetic trees according to the corresponding indices $\mathfrak{C}_{D,f}$ are exactly the fully symmetric ones. Next, for each one of these two functions $f$ and for three popular dissimilarities $D$ (the variance, the standard deviation, and the mean deviation from the median), we determine the range of values of $\mathfrak{C}_{D,f}$ on the sets of phylogenetic trees with a given number $n$ of leaves. We end the paper by assessing the performance of one of these indices on TreeBASE and using it to show that the trees in this database do not seem to follow either the uniform model for multifurcating trees or the $\alpha$-$\gamma$-model, for any values of $\alpha$ and $\gamma$.