the state space of genealogies. We complement this model with a new algorithm for fast likelihood calculations.

Inference for $(N_e(t))_{t\geq 0}$ has important applications in many fields, such as genetics, anthropology, and public health. In the absence of natural selection, the effective population size can be used to approximate census population size. While census population size estimates can be difficult to obtain due to high costs and challenging sampling designs, we can reconstruct past population sizes from observed signatures of genetic diversity in a sample of the population. For example, one can estimate the population size of a virus from genetic samples in a situation where census counts are believed to be inaccurate, as it is common during an epidemic. Inferring population size dynamics – timing of population events, growth and decline rates – rather than estimating census counts, may be of scientific interest. For example, [@sha04] reconstructed bison population dynamics, providing new insights into the extinction of Beringian bison. In this paper, we include two studies supporting the motivations highlighted above: in one study we analyze viral samples of SARS-CoV-2, the virus responsible for the coronavirus disease, and in a second study, we analyze ancient samples of bison in 