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Consider a birth and death process started from one individual in which eachindividual gives birth at rate $\lambda$ and dies at rate $\mu$, so that thepopulation size grows at rate $r = \lambda - \mu$. Lambert and Harris,Johnston, and Roberts came up with methods for constructing the exact genealogyof a sample of size $n$ taken from this population at time $T$. We use theconstruction of Lambert, which is based on the coalescent point process, toobtain asymptotic results for the site frequency spectrum associated with thissample. In the supercritical case $r > 0$, our results extend results ofDurrett for exponentially growing populations. In the critical case $r = 0$,our results parallel those that Dahmer and Kersting obtained for Kingman'scoalescent.

Asymptotics for the site frequency spectrum associated with the genealogy of a birth and death process