Biodiversity conservation requires a framework for prioritising limited resources to the many endangered species. One such framework that has seen much attention and is considered extensively in this thesis, is the Noah's Ark Problem (NAP). The NAP combines a biodiversity measure (Phylogenetic Diversity; PD) with species survival probabilities and conservation costs. The aim of the NAP is to allocate the limited conservation resources such that the future expected PD is maximised. Obtaining optimal solutions to the NAP is a computationally complex problem to which several efficient algorithms are provided here. An extension to the NAP is also developed which allows uncertainty about the survival probability estimates to be included. Using this extension we show that the NAP is robust to uncertainty in these parameters and that even very poor estimates are beneficial. To justify using or promoting PD, it must produce a significant increase in the amount of biodiversity that is preserved. We show that the increase attainable from the NAP is typically around 20% but may be as high as 150%. An alternative approach to PD and the NAP is to prioritise species using simple species specific indices. The benefit of these indices is that they are easy to calculate, explain and integrate into existing management frameworks. Here we investigate the use of such indices and show that they provide between 60% and 80% of the gains obtainable using PD. To explore the expected behaviours of conservation methods (such as the NAP) a distribution of phylogenetics trees is required. Evolutionary models describe the diversification process by which a single species gives rise to multiple species. Such models induce a probability distribution on trees and can therefore be used to investigate the expected behaviour of conservation methods. Even simple and widely used models, such as the Yule model, remain poorly understood. In this thesis we present some new analytic results and methods for sampling trees from a broad range of evolutionary models. Lastly we introduce a new model that provides a simple biological explanation for a long standing discrepancy between models and trees derived from real data -- the tree balance distribution.

Biodiversity conservation requires a framework for prioritising limited resources to the many endangered species. One such framework that has seen much attention and is considered extensively in this thesis, is the Noah's Ark Problem (NAP). The NAP combines a biodiversity measure (Phylogenetic Diversity; PD) with species survival probabilities and conservation costs. The aim of the NAP is to allocate the limited conservation resources such that the future expected PD is maximised. Obtaining optimal solutions to the NAP is a computationally complex problem to which several efficient algorithms are provided here. An extension to the NAP is also developed which allows uncertainty about the survival probability estimates to be included. Using this extension we show that the NAP is robust to uncertainty in these parameters and that even very poor estimates are beneficial. To justify using or promoting PD, it must produce a significant increase in the amount of biodiversity that is preserved. We show that the increase attainable from the NAP is typically around 20% but may be as high as 150%. An alternative approach to PD and the NAP is to prioritise species using simple species specific indices. The benefit of these indices is that they are easy to calculate, explain and integrate into existing management frameworks. Here we investigate the use of such indices and show that they provide between 60% and 80% of the gains obtainable using PD. To explore the expected behaviours of conservation methods (such as the NAP) a distribution of phylogenetics trees is required. Evolutionary models describe the diversification process by which a single species gives rise to multiple species. Such models induce a probability distribution on trees and can therefore be used to investigate the expected behaviour of conservation methods. Even simple and widely used models, such as the Yule model, remain poorly understood. In this thesis we present some new analytic results and methods for sampling trees from a broad range of evolutionary models. Lastly we introduce a new model that provides a simple biological explanation for a long standing discrepancy between models and trees derived from real data -- the tree balance distribution.