We introduce the mid-depth method, a practical approach for testing hypotheses of demographic history using genealogies reconstructed from sequence data. The relative positions of internal nodes within a genealogy contain information about past population dynamics. We explain how this information can be used to (1) test the null hypothesis of constant population size and (2) estimate the growth rate and current population size of an exponentially growing population. Simulation tests indicate that, as expected, estimates of exponential growth rates are sometimes biased. The mid-depth method is computationally rapid and does not require knowledge of the sample's mutation rate. However, it does assume that the reconstructed genealogy is correct and is therefore best suited to the analysis of variation-rich viral data sets. When applied to HIV-1 sequence data, the mid-depth method provides phylogenetic evidence of different exponential growth rates for subtypes A and B. We posit that this difference in growth rate reflects the different transmission routes and epidemiological histories of the two subtypes.