# Tree Height and the Asymptotic Mean of the Colijn–Plazzotta Rank of Unlabeled Binary Rooted Trees

## TL;DR

This paper explores how the rank of a tree structure relates to its height and derives asymptotic properties for different tree models used in mathematical phylogenetics.

## Contribution

The paper introduces new asymptotic results for the Colijn–Plazzotta rank of binary trees under three models, resolving open problems in mathematical phylogenetics.

## Key findings

- The expected value of log log f(τn) grows as 2√(πn) for uniform and labeled binary tree models.
- Under the Yule–Harding model, the expected value of log log f(τn) grows proportionally to α log n, where α ≈ 4.31107.
- The mean rank f(τn) is closely tied to the rank of the highest-ranked caterpillar tree.

## Abstract

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 maximal path length from a leaf to the root. We consider the rank \documentclass[12pt]{minimal}
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\begin{document}$$f(\tau _n)$$\end{document}f(τn) of a random n-leaf tree \documentclass[12pt]{minimal}
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\begin{document}$$\tau _n$$\end{document}τn 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 \documentclass[12pt]{minimal}
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\begin{document}$$\log \log f(\tau _n)$$\end{document}loglogf(τn). In particular, we find \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {E}}\{\log _2 \log f(\tau _n)\} \sim 2 \sqrt{\pi n}$$\end{document}E{log2logf(τn)}∼2πn for uniformly random unlabeled ordered binary rooted trees and uniformly random leaf-labeled binary trees, and for a constant \documentclass[12pt]{minimal}
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\begin{document}$$\alpha \approx 4.31107$$\end{document}α≈4.31107, \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {E}}\{\log _2 \log f(\tau _n)\} \sim \alpha \log n $$\end{document}E{log2logf(τn)}∼αlogn for leaf-labeled binary trees under the Yule–Harding model. We show that the mean of \documentclass[12pt]{minimal}
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\begin{document}$$f(\tau _n)$$\end{document}f(τn) itself under the three models is largely determined by the rank \documentclass[12pt]{minimal}
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\begin{document}$$c_{n-1}$$\end{document}cn-1 of the highest-ranked tree—the caterpillar—obtaining an asymptotic relationship with \documentclass[12pt]{minimal}
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\begin{document}$$\pi _n c_{n-1}$$\end{document}πncn-1, where \documentclass[12pt]{minimal}
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\begin{document}$$\pi _n$$\end{document}πn 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.