Combinatorial properties of phylogenetic diversity indices

  • Kristina Wicke
  • Mike SteelEmail author


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.


Phylogenetic tree Diversity index Shapley value Biodiversity measures 

Mathematics Subject Classification

05-C05 05-C35 91-A12 92-D15 



We thank Arne Mooers for a number of helpful suggestions, and the two anonymous reviewers for detailed comments on an earlier version of this manuscript. We also thank François Bienvenu for pointing out an alternative proof of Lemma 1, and Mareike Fischer for helpful comments concerning Sect. 4. The first author also thanks the German Academic Scholarship Foundation for a doctoral scholarship.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceUniversity of GreifswaldGreifswaldGermany
  2. 2.Biomathematics Research CentreUniversity of CanterburyChristchurchNew Zealand

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