  Tree shape statistics provide valuable quantitative insights into
evolutionary mechanisms underpinning phylogenetic trees, a commonly used graph
representation of evolution systems ranging from viruses to species. By
developing limit theorems for a version of extended P\'olya urn models in which
negative entries are permitted for their replacement matrices, we present
strong laws of large numbers and central limit theorems for asymptotic joint
distributions of two subtree counting statistics, the number of cherries and
that of pitchforks, for random phylogenetic trees generated by two widely used
null tree models: the proportional to distinguishable arrangements (PDA) and
the Yule-Harding-Kingman (YHK) models. Our results indicate that the limiting
behaviour of these two statistics, when appropriately scaled, are independent
of the initial trees used in the tree generating process.
