Counting Spinal Tree-Child Networks via Word Encodings and Generating Functions

TL;DR

This paper presents two combinatorial methods to enumerate spinal tree-child phylogenetic networks, providing explicit formulas and generating functions for their counts.

Contribution

It introduces a novel word model and a symbolic recursive approach to count and analyze spinal tree-child networks.

Findings

Explicit enumeration formulas for spinal tree-child networks.

A closed-form generating function derived from symbolic methods.

Recursive specifications enabling coefficient computation.

Abstract

We study the enumeration of spinal tree-child phylogenetic networks, a rigid family of tree-child networks in which all internal vertices lie on a single root--to--leaf path. We provide two complementary combinatorial frameworks. First, we introduce a word model: unlabeled spinal networks correspond to a suitable class of restricted words with fixed multiplicities, taken modulo a simple relabeling equivalence, which yields an explicit closed enumeration. Second, we develop a symbolic-method approach based on a marked version of trees that admits a clean recursive specification; its boxed-product translation leads to a solvable bivariate generating function and a direct derivation of the coefficients.