# Combinatorial properties of phylogenetic diversity indices

## TL;DR

This paper explores the mathematical properties and relationships of phylogenetic diversity indices, focusing on Fair Proportion and Equal Splits, their equivalence, and extensions to unrooted trees, with implications for evolutionary heritage measurement.

## Contribution

It characterizes when FP and ES indices differ or are identical, and examines their relationship with the Shapley Value on unrooted trees, introducing new analogues.

## Key findings

- FP and ES can differ depending on tree shape
- FP is equivalent to the Shapley Value on rooted trees
- New indices related to Pauplin representation are introduced

## Abstract

Phylogenetic diversity indices provide a formal way to apportion 'evolutionary heritage' across species. Two natural diversity indices are Fair Proportion (FP) and Equal Splits (ES). FP is also called 'evolutionary distinctiveness' and, for rooted trees, is identical to the Shapley Value (SV), which arises from cooperative game theory. In this paper, we investigate the extent to which FP and ES can differ, characterise tree shapes on which the indices are identical, and study the equivalence of FP and SV and its implications in more detail. We also define and investigate analogues of these indices on unrooted trees (where SV was originally defined), including an index that is closely related to the Pauplin representation of phylogenetic diversity.