  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 $1/2<\alpha<1$.
