On cherry and pitchfork distributions of random rooted and unrooted phylogenetic trees

Tree shape statistics are important for investigating evolutionary mechanisms mediating phylogenetic trees. As a step towards bridging shape statistics between rooted and unrooted trees, we present a comparison study on two subtree statistics known as numbers of cherries and pitchforks for the proportional to distinguishable arrangements (PDA) and the Yule-Harding-Kingman (YHK) models. Based on recursive formulas on the joint distribution of the number of cherries and that of pitchforks, it is shown that cherry distributions are log-concave for both rooted and unrooted trees under these two models. Furthermore, the mean number of cherries and that of pitchforks for unrooted trees converge respectively to those for rooted trees under the YHK model while there exists a limiting gap of 1∕4 for the PDA model. Finally, the total variation distances between the cherry distributions of rooted and those of unrooted trees converge for both models. Our results indicate that caution is required for conducting statistical analysis for tree shapes involving both rooted and unrooted trees.