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Using Phylogenetic Analysis to Investigate Eukaryotic Gene Origin

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Published on: August 14, 2018

Enumeration of rooted binary perfect phylogenies.

Chloe E Shiff¹, Noah A Rosenberg²

¹Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305, USA.

Discrete Applied Mathematics (Amsterdam, Netherlands : 1988)

|December 18, 2025

Summary

This study enumerates rooted binary perfect phylogenies, a generalization of unlabeled trees used in biology. These phylogenies grow faster than unlabeled trees, with specific counts derived for various leaf numbers and shapes like caterpillars.

Keywords:

05A16 05C05 92D15 MSC2020 classification: 05A15 generating functions perfect phylogenies unlabeled trees

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Last Updated: Jan 8, 2026

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A Concoction Pipeline for Generating Molecular Operational Taxonomic Units (MOTUs) Among Riparian and Aquatic Beetles

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Published on: July 11, 2025

Area of Science:

Combinatorics and Graph Theory
Computational Biology
Phylogenetic Analysis

Background:

Rooted binary perfect phylogenies generalize rooted binary unlabeled trees.
Leaf values represent counts of indistinguishable lineages in biological contexts.
Standard rooted binary unlabeled trees have leaf values of 1.

Purpose of the Study:

To enumerate rooted binary perfect phylogenies with n leaves and sample size s.
To analyze the growth rate and distribution of lineages across leaves.
To provide mathematical frameworks for understanding phylogenetic structures.

Main Methods:

Recursive enumeration of phylogenies based on sample size and number of leaves.
Derivation of generating functions to count phylogenies.
Asymptotic analysis to determine growth rates and distribution patterns.

Main Results:

Asymptotic growth rate for phylogenies with sample size s is ≈ 0.3519(3.2599)^s * s^(-3/2).
Closed-form counts derived for phylogenies with 2, 3, and 4 leaves.
Generating functions and exact counts (2^s / 3) found for caterpillar phylogenies.

Conclusions:

Rooted binary perfect phylogenies exhibit faster growth than unlabeled trees.
Mathematical models provide insights into the structure and distribution of lineages.
These enumerations offer valuable tools for biological and phylogenetic studies.