  The Yule-Harding-Kingman (YHK) model and the proportional to distinguishable
arrangements (PDA) model are two binary tree generating models that are widely
used in evolutionary biology. Understanding the distributions of clade sizes
under these two models provides valuable insights into macro-evolutionary
processes, and is important in hypothesis testing and Bayesian analyses in
phylogenetics. Here we show that these distributions are log-convex, which
implies that very large clades or very small clades are more likely to occur
under these two models. Moreover, we prove that there exists a critical value
$\kappa(n)$ for each $n\geqslant 4$ such that for a given clade with size $k$,
the probability that this clade is contained in a random tree with $n$ leaves
generated under the YHK model is higher than that under the PDA model if
$1<k<\kappa(n)$, and lower if $\kappa(n)<k<n$. Finally, we extend our results
to binary unrooted trees, and obtain similar results for the distributions of
clan sizes.
