Yule-generated trees constrained by node imbalance

TL;DR

This paper introduces Omega-trees, a subclass of Yule-generated trees constrained by node imbalance, and analyzes their combinatorial properties, revealing that some statistical measures remain unchanged even with imbalance constraints.

Contribution

The paper defines Omega-trees based on imbalance parameter w and studies their properties, showing invariance in certain statistics despite constraints.

Findings

Expectation and variance of the number of two-leaf subtrees match unconstrained trees for small w.

Omega-trees maintain several statistical invariants despite reduced space.

Imbalance parameter w influences the likelihood of tree types under neutral evolution models.

Abstract

The Yule process generates a class of binary trees which is fundamental to population genetic models and other applications in evolutionary biology. In this paper, we introduce a family of sub-classes of ranked trees, called Omega-trees, which are characterized by imbalance of internal nodes. The degree of imbalance is defined by an integer 0 <= w. For caterpillars, the extreme case of unbalanced trees, w = 0. Under models of neutral evolution, for instance the Yule model, trees with small w are unlikely to occur by chance. Indeed, imbalance can be a signature of permanent selection pressure, such as observable in the genealogies of certain pathogens. From a mathematical point of view it is interesting to observe that the space of Omega-trees maintains several statistical invariants although it is drastically reduced in size compared to the space of unconstrained Yule trees. Using generating functions, we study here some basic combinatorial properties of Omega-trees. We focus on the distribution of the number of subtrees with two leaves. We show that expectation and variance of this distribution match those for unconstrained trees already for very small values of w.