Phylogenetic diversity (PD) is a popular measure of biodiversity, with particular applications to conservation management. It brings a focus to the evolutionary relationships between species that is missing in simpler approaches, such as the use of species richness. PD attains this focus by considering species in terms of their positions on a phylogenetic tree. The mathematical properties of PD, and a suite of methods derived from it, have been studied since its introduction in the early 1990’s. In this thesis, we explore these properties further, covering three main aspects of PD-related studies.

The first strand of the thesis covers the study and comparison of the PD values of sets of a fixed size. We use combinatorial and algorithmic approaches to understand those sets of species that obtain the extreme PD scores for sets of their size. A combinatorial characterisation of maximum PD sets is provided. This leads to a polynomial-time algorithm for calculating the number of maximum PD sets of each size by applying a generating function. We then use this characterisation to maximise a linear function on the leaves of a phylogenetic tree, subject to the solution being a maximum PD set. Additionally, dynamic programming is used to find solutions to the dual problem, determining minimum PD sets of each size.

The second strand involves phylogenetic diversity indices, a type of function that partitions the PD of a set of species among its constituent members. We give a formal definition of this class of function, and investigate the properties of functions in this class. This process is aided by a description of diversity indices as points within a convex space, whose dimension and extremal points we describe. Particularly, we show that rankings derived from these measures are susceptible to being disrupted by the extinction of some of the species being measured. We introduce a number of new measures that avoid this disruption to a greater extent than existing approaches.

The third strand deals with the link between PD and feature diversity (FD), another means of measuring biodiversity. We provide models for evolution of features on phylogenetic trees that account for loss of features, such as the loss of flight in some bird species. Doing so leads to results showing that PD is an imperfect proxy for FD unless feature loss is (unrealistically) ignored. We show how our new measure, EvoHeritage, spans a continuum that connects PD and SR at the extremes, based on the rate of assumed feature loss.

The distinct parts of this thesis are linked by an aim to better understand what is meant by the concept of biodiversity and to investigate how that understanding is reflected in the way that we measure this idea. We provide a mathematical approach, complemented by a number of algorithms that enable these ideas to be put into practice.