  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.
