Such a bimodal distribution is the characteristic signature of genealogies in populations with a relatively constant Ne in the past. The bimodal pattern is caused by roughly equal times to coalescence of all internal and external branches. In contrast, populations that had rapidly growing or shrinking Ne in the past tend to have distinct mismatch distributions. In populations that have rapidly growing Ne, most coalescence events happen early in the genealogy near the MRCA since the probability of coalescence decreases toward the present (see the left-hand genealogy in Fig. 8.20). This leads to long external branches that each experience many unique mutations. The mismatch distribution then has a high frequency of sequence pairs with a high degree of mismatch and few sequence pairs with a low degree of mismatch. Alternatively, populations that experienced continual declines in Ne have genealogies where most coalescence events happen near the present because the probability of coalescence increases toward the present (see the right-hand genealogy in Fig. 8.20). In a shrinking population, the mismatch distribution tends to have a high frequency of sequence pairs with low mismatch counts.

A related way to view polymorphism is by examining the distribution of haplotype frequencies in a sample of sequences. Such haplotype frequency distributions show the proportion of sequences in a population that represent each of the observed sequence alleles (assuming that individuals are haploid or homozygous). Under neutrality and constant effective population size (see Fig. 8.20) a range of haplotype frequencies are expected from very frequent to rare. When populations are growing rapidly or there is balancing selection, there is expected to be an excess of rare haplotypes produced by the excess length of external branches in the genealogy. When populations are shrinking rapidly or there is

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