  Many population genetic models have been developed for the purpose of
inferring population size and growth rates from random samples of genetic data.
We examine two popular approaches to this problem, the coalescent and the
birth-death-sampling model, in the context of estimating population size and
birth rates in a population growing exponentially according to the birth-death
branching process. For sequences sampled at a single time, we found the
coalescent and the birth-death-sampling model gave virtually indistinguishable
results in terms of the growth rates and fraction of the population sampled,
even when sampling from a small population. For sequences sampled at multiple
time points, we find that the birth-death model estimators are subject to large
bias if the sampling process is misspecified. Since birth-death-sampling models
incorporate a model of the sampling process, we show how much of the
statistical power of birth-death-sampling models arises from the sequence of
sample times and not from the genealogical tree. This motivates the development
of a new coalescent estimator, which is augmented with a model of the known
sampling process and is potentially more precise than the coalescent that does
not use sample time information.
