Distributions of cherries and pitchforks for the Ford model

Distributional properties of tree shape statistics under random phylogenetic tree models play an important role in investigating evolutionary forces underlying real world phylogenies. In this paper, we study two subtree counting statistics, the number of cherries and that of pitchforks for Ford's alpha model, a one-parameter family of random phylogenetic tree models which includes as specific instances of both the uniform and the Yule models, two tree models commonly used in phylogenetics. Based on a version of the extended P\'olya urn models, in which negative entries are permitted for their replacement matrices, we obtain the strong laws of large numbers and the central limit theorems for the joint distribution of these two count statistics for the Ford model. Furthermore, we derive a recursive formula for computing the exact joint distribution of these two statistics, which leads to higher order asymptotic expansions of their marginal and joint moments.