The formulation for the rarefaction of Phylogenetic Diversity (PD) is given in expanded form to show its simplicity and its connection to the classic formula for the rarefaction of species richness (Hurlbert 1971; Simberloff 1972). The method is exact and efficient and should be preferred over the algorithmic (Monte Carlo) solution involving repeated random sub-sampling. Further, the extension to the calculation of ∆PD provides a flexible and general framework for the measurement of biodiversity as phylogenetic evenness, phylogenetic beta-diversity or phylogenetic dispersion. The applications of PD rarefaction and ∆PD presented here are hopefully useful in improving understanding of the importance of rarefaction in ecology and in guiding future applications of the method. There are, I believe, exciting prospects for PD rarefaction in the future, including as a general method for standardising PD by removing variation with species richness, and for predicting unseen (i.e. un-sampled) PD. The recent availability of comprehensive phylogenies (Bininda-Emonds et al. 2007; Jetz et al. 2012) and rich data on species occurrences (Flemons et al. 2007), coupled with analytical advances such as PD rarefaction, allows us to better understand the distribution of Phylogenetic Diversity on the surface of the Earth and the processes giving rise to that distribution. This is valuable for its own sake but will also inform efforts to conserve as much of the Tree of Life as possible in the face of future extinctions (Rosauer and Mooers 2013).

Acknowledgements I would like to thank Frederick Matsen (Fred Hutchinson Cancer Research Center), Daniel Faith (Australian Museum), Peter Wilson (Macquarie University), Anne Chao (National Tsing Hua University) and Robert Colwell (University of Connecticut) for their input regarding the rarefaction of PD and what it all might mean. Daniel Miranda-Esquivel (Universidad Industrial de Santander) and Pedro Cardoso (University of Helsinki) provided helpful feedback on an earlier version of this chapter. This research was supported by the Australian Research Council (DP0665761 and DP1095200).

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