Enumeration of rooted binary perfect phylogenies

TL;DR

This paper explores the enumeration of rooted binary perfect phylogenies, generalizing rooted binary unlabeled trees by incorporating positive integer counts at leaves, with applications in biological lineage analysis.

Contribution

It introduces enumeration methods for rooted binary perfect phylogenies, extending classical tree enumeration to include leaf counts for biological data modeling.

Findings

Derived enumeration formulas for rooted binary perfect phylogenies

Characterized the relationship between perfect phylogenies and unlabeled trees

Provided combinatorial insights applicable to biological lineage studies

Abstract

Rooted binary perfect phylogenies provide a generalization of rooted binary unlabeled trees in which each leaf is assigned a positive integer value that corresponds in a biological setting to the count of the number of indistinguishable lineages associated with the leaf. For the rooted binary unlabeled trees, these integers equal 1. We address a variety of enumerative problems concerning rooted binary perfect phylogenies with sample size $s$: the rooted binary unlabeled trees in which a sample of size $s$ lineages is distributed across the leaves of an unlabeled tree with $n$ leaves, $1 \leq n \leq s$. The enumerations further characterize the rooted binary perfect phylogenies, which include the rooted binary unlabeled trees, and which can provide a set of structures useful for various biological contexts.