Replication Principle for Phylogenetic Diversity Measures

The replication principle was generalized to a phylogenetic version in Chao et al. (2010). Suppose there are N equally large and completely phylogenetically distinct assemblages (no shared lineages across assemblages, though lineages within an assemblage may be shared); see Fig. 2 (reproduced from Chiu et al. 2014) for an illustrative example. Suppose these assemblages have the same phylogenetic Hill number X. If these assemblages are pooled, then the pooled assemblages must have a phylogenetic Hill number N × X. In the proof of this replication principle, Chao et al. (2010) assumed that these N assemblages have the same mean branch lengths. Here we relax this assumption and allow assemblages to have different mean branch lengths. (In the special case of ultrametric trees, this means that we allow different time perspectives for different assemblages.)

Suppose in assemblage k, the mean branch length is Tk , and the branch set is (we omit Tk in the subscript and just use Bk in the following proof for notational simplicity) with branch lengths {Lik; i∈Bk} and the corresponding nodes

Fig. 2 Replication Principle for two completely phylogenetically distinct assemblages with totally different structures. Left panel: Assemblage 1 (black) includes three species with species relative abundances {p11, p21, p31} for the three tips. Assemblage 2 (grey) includes four species with species relative abundances {p12, p22, p32, p42} for the four tips. The diversity of the pooled tree is double of that of each tree as long as the two assemblages are completely phylogenetically distinct as shown (no lineages shared between assemblages, though lineages within an assemblage may be shared) and have identical mean diversities (i.e., phylogenetic Hill number). Right panel: The same is valid for two completely phylogenetically distinct non-ultrametric assemblages (This figure is reproduced from Fig. 1 of Chiu et al. 2014)

abundances {aik; i∈Bk}, k = 1, 2, …, N. Assume that all assemblages have the same

phylogenetic Hill numbers q implying abundance aik in the pooled tree becomes aik/N, and the mean branch length becomes T = (1 / N ) åTk . Then the phylogenetic Hill number of order q for the pooled assemblage becomes L aq for all k =1,

D (Ti2, …, N. When the N trees are pooled with equal weight for each tree, each node

This proves a stronger version of the replication principle for phylogenetic Hill numbers. Note the mean branch length in the pooled assemblage is the average of

individual mean branch lengths. For example, if q D (T = 2) = q D (T = 6) = 10,

then in an effective sense, there are ten lineages with mean branch length 2 in

Assemblage 1 and there are ten lineages with mean branch length 6 in Assemblage

2. The replication principle implies that there are 20 lineages in the pooled tree with mean branch length 4. Since q PD (T ) = q D (T )´T , the replication principle for

k k k

the phylogenetic diversity q PD (T ) does need the assumption that all assemblages have the same mean branch lengths (T1 = T2 = ¼= TN ) . The proof is parallel and

thus omitted.