Distributions of cherries and pitchforks for the Ford model

We study two fringe 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 that includes the uniform and the Yule models, two tree models commonly used in phylogenetics. Based on a nonuniform version of the extended P\'olya urn models in which negative entries are permitted for their replacement matrices, we obtain the strong law of large numbers and the central limit theorem 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. This leads to exact formulas for their means and higher order asymptotic expansions of their second moments, which allows us to identify a critical parameter value for the correlation between these two statistics. That is, when $n$ is sufficiently large, they are negatively correlated for $0\le \alpha \le 1/2$ and positively correlated for $1/2<\alpha<1$.