Combinatorial properties of phylogenetic diversity indices

Area of Science:

• Ecology
• Evolutionary Biology
• Phylogenetics

Background:

• Phylogenetic diversity indices quantify evolutionary heritage across species.
• Fair Proportion (FP), also known as evolutionary distinctiveness, and Equal Splits (ES) are two key indices.
• FP is equivalent to Shapley Value (SV) for rooted phylogenetic trees.

Purpose of the Study:

• To investigate the differences between FP and ES phylogenetic diversity indices.
• To characterize tree shapes where FP and ES yield identical results.
• To further analyze the equivalence of FP and SV and its implications, including on unrooted trees.

Main Methods:

• Comparative analysis of FP and ES indices on different phylogenetic tree structures.
• Characterization of tree topologies that result in equivalent FP and ES values.
• Investigation of FP and SV equivalence, including analogues for unrooted trees.

Main Results:

• FP and ES indices can diverge significantly depending on tree shape.
• Specific tree shapes lead to the identity of FP and ES.
• The equivalence between FP and SV is confirmed and extended to unrooted tree contexts.

Conclusions:

• Understanding the distinctions and equivalences between phylogenetic diversity indices is crucial for accurate evolutionary heritage assessment.
• The study provides insights into the mathematical properties of FP, ES, and SV across diverse phylogenetic structures.
• New analogues of these indices are proposed for unrooted phylogenetic trees, expanding their applicability.