Relationship between our results and an effective population-size approximation. (A) A typical coalescent tree in a neutral population of constant size. The coalescent probability per generation between a random pair of individuals is the inverse population size. Time runs from the past at the top to the present at the bottom. (B) An example of a neutral coalescent tree in a population that was smaller in the past than the present. The population size is shown as the width in green. Coalescence events are more likely to occur when the population size is smaller. (C) The effective population-size history for an individual experiencing purifying selection according to our model. The individual spends on average

generations in class

*k*, which has a total size

*Nh*_{k}. Note that pairs of individuals are sampled from different classes

*k* (

*i.e.*, they are not all sampled from the bottom of this picture). Further, the coalescence probabilities also include a factor of

*A*/2, which reflects the probability that two lineages are in the same class at the same time. (D) The historically varying effective population size

*N*_{e}(

*t*) for a pair of individuals sampled from classes

*k* and

*k*′, as defined in the text, for several values of

*k* and

*k*′. The

*N*_{e}(

*t*) for two individuals sampled at random from the whole population is also shown. Here

*N* = 5 × 10

^{4},

*U*_{d}/

*s* = 6, and

*s* = 10

^{−3}.