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.

中文翻译：

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关于随机生根和未生根的系统发育树的樱桃和干草叉分布。

树形统计对于调查介导系统发育树的进化机制很重要。为了在有根和无根的树之间架起形状统计的一步，我们对两个子树统计进行了比较研究，这两个树分别是樱桃和干草叉的比例可分比例排列（PDA）和Yule-Harding-Kingman（YHK）模型。基于关于樱桃和干草叉数量的联合分布的递归公式，表明在这两种模型下，对于有根和无根树，樱桃的分布都是对数凹形的。此外，在YHK模型下，无根树的平均樱桃数和干草叉分别收敛于有根树的平均数，而PDA模型的极限缺口为1∕4。最后，两种模型中，有根和无根树的樱桃分布之间的总变化距离收敛。我们的结果表明，对涉及有根和无根树木的树木形状进行统计分析时需要谨慎。