Tree height and the asymptotic mean of the Colijn-Plazzotta rank of unlabeled binary rooted trees

The Colijn–Plazzotta ranking is a bijective encoding of the unlabeled binary rooted trees with positive integers. We show that the rank $f(t)$ of a tree $t$ is closely related to its height $h$, the length of the longest path from a leaf to the root. We consider the rank $f({\tau }_n)$ of a random $n$-leaf tree ${\tau }_n$ under each of three models: (i) uniformly random unlabeled unordered binary rooted trees, or unlabeled topologies; (ii) uniformly random leaf-labeled binary trees, or labeled topologies under the uniform model; and (iii) random binary search trees, or labeled topologies under the Yule–Harding model. Relying on the close relationship between tree rank and tree height, we obtain results concerning the asymptotic properties of $\log \log f({\tau }_n)$. In particular, we find $\mathbb{E}\{{\log }_2\log f({\tau }_n)\}\sim 2\sqrt{\pi n}$ for uniformly random unlabeled ordered binary rooted trees and uniformly random leaf-labeled binary trees, and for a constant $\alpha \approx 4.31107$, $\mathbb{E}\{{\log }_2\log f({\tau }_n)\}\sim \alpha \log n$ for leaf-labeled binary trees under the Yule–Harding model. We show that the mean of $f({\tau }_n)$ itself under the three models is largely determined by the rank $c_{n-1}$ of the highest-ranked tree – the caterpillar – obtaining an asymptotic relationship with ${\pi }_n{c}_{n-1}$, where ${\pi }_n$ is a model-specific function of $n$. The results resolve open problems, providing a new class of results on an encoding useful in mathematical phylogenetics.