Exact enumeration of cherries and pitchforks in ranked trees under the coalescent model

TL;DR

This paper develops a new generating function approach to exactly enumerate and analyze the probabilities of cherries and pitchforks in ranked trees generated by the coalescent model, providing recursive formulas for their joint distributions.

Contribution

It introduces a novel method based on generating functions for exact enumeration and probabilistic analysis of cherries and pitchforks in ranked trees under the coalescent model.

Findings

Derived exact probabilities for cherries in fixed-size ranked trees

Extended method to include pitchforks in the analysis

Established recursive formulas for joint and conditional probabilities

Abstract

We consider exact enumerations and probabilistic properties of ranked trees when generated under the random coalescent process. Using a new approach, based on generating functions, we derive several statistics such as the exact probability of finding k cherries in a ranked tree of fixed size n. We then extend our method to consider also the number of pitchforks. We find a recursive formula to calculate the joint and conditional probabilities of cherries and pitch- forks when the size of the tree is fixed.