Sound Colless-like balance indices for multifurcating trees.

Area of Science:

• Phylogenetics
• Computational Biology
• Tree Theory

Background:

• The Colless index is a standard balance measure for bifurcating phylogenetic trees.
• This index is not applicable to multifurcating phylogenetic trees, limiting balance assessment in broader evolutionary contexts.

Purpose of the Study:

• To generalize the Colless index for multifurcating phylogenetic trees.
• To introduce a family of Colless-like balance indices, denoted as [Formula: see text], adaptable to various dissimilarities and weight functions.
• To identify sound Colless-like indices and analyze the balance properties of multifurcating trees.

Main Methods:

• Proposed a family of Colless-like indices [Formula: see text] defined by a dissimilarity D and a weight function [Formula: see text].
• Proved the soundness of [Formula: see text] for specific weight functions f(n) = ln(n + e) and f(n) = en.
• Analyzed the most unbalanced trees for these sound indices across different dissimilarities and leaf numbers (n).
• Developed the R package "CollessLike" for computing and comparing these indices.

Main Results:

• Identified sound Colless-like indices using logarithmic and exponential weight functions.
• Demonstrated that the growth pace of the weight function influences the definition of tree balance.
• Found that trees in TreeBASE do not conform to the uniform or α-γ models for multifurcating trees.

Conclusions:

• The proposed Colless-like indices provide a robust framework for assessing balance in multifurcating phylogenetic trees.
• The choice of weight function significantly impacts the interpretation of tree balance.
• Empirical analysis suggests that real-world phylogenetic data may deviate from common theoretical models.