  Phylogenetic diversity indices are commonly used to rank the elements in a
collection of species or populations for conservation purposes. The derivation
of these indices is typically based on some quantitative description of the
evolutionary history of the species in question, which is often given in terms
of a phylogenetic tree. Both rooted and unrooted phylogenetic trees can be
employed, and there are close connections between the indices that are derived
in these two different ways. In this paper, we introduce more general
phylogenetic diversity indices that can be derived from collections of subsets
(clusters) and collections of bipartitions (splits) of the given set of
species. Such indices could be useful, for example, in case there is some
uncertainty in the topology of the tree being used to derive a phylogenetic
diversity index. As well as characterizing some of the indices that we
introduce in terms of their special properties, we provide a link between
cluster-based and split-based phylogenetic diversity indices that uses a
discrete analogue of the classical link between affine and projective geometry.
This provides a unified framework for many of the various phylogenetic
diversity indices used in the literature based on rooted and unrooted
phylogenetic trees, generalizations and new proofs for previous results
concerning tree-based indices, and a way to define some new phylogenetic
diversity indices that naturally arise as affine or projective variants of each
other.
