Biodiversity Conservation and Phylogenetic Systematics

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expected PD of Faith ( 2013 ) but differs in that random draws are without replace-
ment following the hypergeometric distribution.

EPDL

No

m

N

m

m

j

T

j

j

[]=×−

 −



































































∑^1









(6)

Finally, it is now possible to calculate the expected PD for a given number of
species. A species, in this context, is simply a collection of individuals in much the
same way as a sample is a collection of individuals, and the same equations apply.

001135

Abundance of species i

000005

Abundance of species i

001112

Abundance of species i

m= 10

m= 5

Fig. 2 An illustration of the process of rarefying Phylogenetic Diversity (PD) by units of individu-
als. An initial sample of ten individuals (m=10)distributedamongfourtips(species)israrefiedto
a subset of five individuals (m=5)byaprocessofrandomsamplingwithoutreplacement.Forthe
rarefiedsamples, 2 ofthe 252 possiblesubsetsareshown.TheexpectedPDunderrarefactionisthe
average sum of branch lengths represented by each of these distinct subsets. The branch lengths
summed to calculate PD are black while those not represented (and thus not summed) are grey.
Note that the rooted definition of PD is used where the path length to the root is always included,
even in the case where only a single tip is represented

The Rarefaction of Phylogenetic Diversity: Formulation, Extension and Application