Reconstructing the genealogy of a sample of alleles

Specifically, let's keep track of the genealogy of alleles. In a finite population, two randomly chosen alleles will be identical by descent with respect to the immediately preceding generation with probability $1/2N_e$. That means that there's a chance that two alleles in generation $t$ are copies of the same allele in generation $t-1$. If the population size is constant, meaning that the number of alleles in the population is remaining constant, then there's also a chance that some alleles present in generation $t-1$ will not have descendants in generation $t$. Looking backward, then, the number of alleles in generation $t-1$ that have descendants in generation $t$ is always less than or equal to the number of alleles in generation $t$. That means if we trace the ancestry of alleles in a sample back far enough, all of them will be descended from a single common ancestor. Figure 1 provides a simple schematic illustrating how this might happen.

Figure 1: A schematic depiction of one possible realization of the coalescent process in a population with 18 haploid gametes. There are four coalescent events in the generation immediately preceding the last one illustrated, one involving three alleles.

Now take a look at Figure 1. Time runs from the top of the figure to the bottom, i.e., the current generation is represented by the circles in the botton row of the figure. Each circle represents an allele. The eighteen alleles in our current sample are descended from only four alleles that were present in the populations ten generations ago. The other fourteen alleles present in the population ten generations ago left no descendants. How far back in time we'd have to go before all alleles are descended from a single common ancestor depends on the effective size of the population, and how frequently two (or more) alleles are descended from the same allele in the preceding generation depends on the effective size of the population, too. But in any finite population the pattern will look something like the one I've illustrated here.