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Tree shape statistics provide valuable quantitative insights intoevolutionary mechanisms underpinning phylogenetic trees, a commonly used graphrepresentation of evolution systems ranging from viruses to species. Bydeveloping limit theorems for a version of extended P\'olya urn models in whichnegative entries are permitted for their replacement matrices, we presentstrong laws of large numbers and central limit theorems for asymptotic jointdistributions of two subtree counting statistics, the number of cherries andthat of pitchforks, for random phylogenetic trees generated by two widely usednull tree models: the proportional to distinguishable arrangements (PDA) andthe Yule-Harding-Kingman (YHK) models. Our results indicate that the limitingbehaviour of these two statistics, when appropriately scaled, are independentof the initial trees used in the tree generating process.

On asymptotic joint distributions of cherries and pitchforks for random phylogenetic trees