The distribution of clade size in a coalescent diversification model

Characterizing the patterns and determinants of biological diversity is a central aim of evolutionary biology. Doing so requires developing expectations for clade size, the relationship between the number of species in a clade and its age, for standard diversification models, such as the coalescent or birth-death process. These expectations are necessary for identifying diversity outliers, specious or depauperate clades in the macroevolutionary context or transmission clusters of particular or little public health concern in the epidemiological context, and for testing alternative diversification hypotheses. Here, we derive a closed-form expression for the distribution of clade sizes under the widely used Kingman coalescent diversification model and extend the results to allow the number of niches (the effective population size) to vary through time. This result complements analogous results for the birth-death model in that it provides expectations for the joint distribution of clade sizes for a case where diversification is strongly density-dependent.