Inference of Historical Population-Size Changes with Allele-Frequency Data. 
With up to millions of nearly neutral polymorphisms now being routinely sampled in population-genomic surveys, it is possible to estimate the site-frequency spectrum of such sites with high precision.  Each frequency class reflects a mixture of potentially unique demographic histories, which can be revealed using theory for the probability distributions of the starting and ending points of branch segments over all possible coalescence trees.  Such distributions are completely independent of past population history, which only influences the segment lengths, providing the basis for estimating average population sizes separating tree-wide coalescence events.  The history of population-size change experienced by a sample of polymorphisms can then be dissected in a model-flexible fashion, and extension of this theory allows estimation of the mean and full distribution of long-term effective population sizes and ages of alleles of specific frequencies.  Here, we outline the basic theory underlying the conceptual approach, develop and test an efficient statistical procedure for parameter estimation, and apply this to multiple population-genomic datasets for the microcrustacean Daphnia pulex.