Distributions of cherries for two models of trees

Null models for generating binary phylogenetic trees are useful for testing evolutionary hypotheses and reconstructing phylogenies. We consider two such null models – the Yule and uniform models – and in particular the induced distribution they generate on the number Cn of cherries in the tree, where a cherry is a pair of leaves each of which is adjacent to a common ancestor. By realizing the process of cherry formation in these two models by extended Polya urn models we show that Cn is asymptotically normal. We also give exact formulas for the mean and standard deviation of the Cn in these two models. This allows simple statistical tests for the Yule and uniform null hypotheses.