Counting, grafting and evolving binary trees

TL;DR

This paper explores the combinatorial properties and probabilistic models of binary trees, focusing on their generation processes like Yule trees, and discusses their structural differences and enumeration.

Contribution

It analyzes how different tree classes and generation processes affect the structural properties and probability distributions of binary trees.

Findings

Ordered Yule trees have closed-form enumeration and simple probability distributions.

Unordered Yule trees lack closed-form enumeration and simple distributions.

The generation process influences the structural and probabilistic properties of binary trees.

Abstract

Binary trees are fundamental objects in models of evolutionary biology and population genetics. Here, we discuss some of their combinatorial and structural properties as they depend on the tree class considered. Furthermore, the process by which trees are generated determines the probability distribution in tree space. Yule trees, for instance, are generated by a pure birth process. When considered as unordered, they have neither a closed-form enumeration nor a simple probability distribution. But their ordered siblings have both. They present the object of choice when studying tree structure in the framework of evolving genealogies.