  In conservation biology, phylogenetic diversity (PD) provides a way to
quantify the impact of the current rapid extinction of species on the
evolutionary `Tree of Life'. This approach recognises that extinction not only
removes species but also the branches of the tree on which unique features
shared by the extinct species arose. In this paper, we investigate three
questions that are relevant to PD. The first asks how many sets of species of
given size $k$ preserve the maximum possible amount of PD in a given tree. The
number of such maximum PD sets can be very large, even for moderate-sized
phylogenies. We provide a combinatorial characterisation of maximum PD sets,
focusing on the setting where the branch lengths are ultrametric (e.g.
proportional to time). This leads to a polynomial-time algorithm for
calculating the number of maximum PD sets of size $k$ by applying a generating
function; we also investigate the types of tree shapes that harbour the most
(or fewest) maximum PD sets of size $k$. Our second question concerns
optimising a linear function on the species (regarded as leaves of the
phylogenetic tree) across all the maximum PD sets of a given size. Using the
characterisation result from the first question, we show how this optimisation
problem can be solved in polynomial time, even though the number of maximum PD
sets can grow exponentially. Our third question considers a dual problem: If
$k$ species were to become extinct, then what is the largest possible {\em
loss} of PD in the resulting tree? For this question, we describe a
polynomial-time solution based on dynamical programming.
