Clade size distributions under the coalescent diversification model

Characterizing patterns of biological diversity is a central goal of evolutionary biology. This requires understanding expectations for clade size-the relationship between the number of species in a clade and its age. Such expectations are key for identifying diversity outliers (e.g., specious or depauperate clades in macroevolution, or unusually large or small transmission clusters in epidemiology) and for testing alternative hypotheses about diversification. Here, we develop a general method for deriving closed-form expressions for the (joint) distribution of clade sizes under a given diversification model. We apply this approach to the constant- and variable-size coalescent model as well as the Yule model. Our results reveal that while the coalescent and Yule models produce qualitatively similar clade size patterns, they exhibit quantitative differences. Leveraging the flexibility of the coalescent framework, we further examine how transmission cluster size distributions differ between rapidly and slowly growing epidemics, finding - counterintuitively - that slowly growing epidemics are more likely to generate large clusters, a pattern often attributed to increased transmission.