Clade size distribution under neutral evolutionary models

Given a labeled tree topology $t$, consider a population $P$ of $k$ leaves chosen among those of $t$. The clade of $P$ is the minimal subtree of $t$ containing $P$ and its size is given by the number of leaves in the clade. When $t$ is selected under the Yule or uniform distribution among the labeled topologies of size $n$, we study the “clade size” random variable determining closed formulas for its probability mass function, its mean, and its variance. Our calculations show that for large $n$ the clade size tends to be smaller under the uniform model than under the Yule model, with a larger variability in the first scenario for values of $k\ge 5$. We apply our probability formulas to investigate set-theoretic relationships between the clades of two populations in a random tree, determining how likely one clade is contained in or it is equal to the other. Our study relates to earlier calculations for the probability that under the Yule model the clade size of $P$ equals the size of $P$ – that is, the population $P$ forms a monophyletic group – and extends known results for the probability that the minimal (non-trivial) clade containing a random taxon has a given size.