Computing tree size under dynamical models of diversification

A phylogenetic tree has three types of attributes: size, shape (topology),
and branch lengths. Phylodynamic studies are often motivated by questions
regarding the size of clades, nevertheless, nearly all of the inference
methods only make use of the other two attributes. In this paper, we ask
whether there is additional information if we consider tree size more
explicitly in phylodynamic inference methods. To address this question, we
first needed to be able to compute the expected tree size distribution
under a specified phylodynamic model; perhaps surprisingly, there is not a
general method for doing so – it is known what this is under a Yule or
constant rate birth-death model but not for the more complicated scenarios
researchers are often interested in. We present three different solutions
to this problem: using i) the deterministic limit; ii) master equations;
and iii) an ensemble moment approximation. Using simulations, we evaluate
the accuracy of these three approaches under a variety of scenarios and
alternative measures of tree size (i.e., sampling through time or only at
the present; sampling ancestors or not). We then use the most accurate
measures for the situation, to investigate the added informational content
of tree size. We find that for two critical phylodynamic questions – i) is
diversification diversity dependent? and, ii) can we distinguish between
alternative diversification scenarios? – knowing the expected tree size
distribution under the specified scenario provides insights that could not
be gleaned from considering the expected shape and branch lengths alone.
The contribution of this paper is both a novel set of methods for
computing tree size distributions and a path forward for richer
phylodynamic inference into the evolutionary and epidemiological processes
that shape lineage trees.