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Many population genetic models have been developed for the purpose ofinferring population size and growth rates from random samples of genetic data.We examine two popular approaches to this problem, the coalescent and thebirth-death-sampling model, in the context of estimating population size andbirth rates in a population growing exponentially according to the birth-deathbranching process. For sequences sampled at a single time, we found thecoalescent and the birth-death-sampling model gave virtually indistinguishableresults in terms of the growth rates and fraction of the population sampled,even when sampling from a small population. For sequences sampled at multipletime points, we find that the birth-death model estimators are subject to largebias if the sampling process is misspecified. Since birth-death-sampling modelsincorporate a model of the sampling process, we show how much of thestatistical power of birth-death-sampling models arises from the sequence ofsample times and not from the genealogical tree. This motivates the developmentof a new coalescent estimator, which is augmented with a model of the knownsampling process and is potentially more precise than the coalescent that doesnot use sample time information.

Sampling through time and phylodynamic inference with coalescent and birth-death models