 Solution

For a haploid population with $N$ possible parents, each pair of ancestral lineages merges backward at rate $1/N$ in the Kingman's coalescent approximation. With $k$ lineages there are $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ pairs. Hence the mean phylogenetic epoch duration isThus $E_2=N$, $E_3=N/3$ and $E_4=N/6$. For an autosomal sample from $N$ diploid individuals, replace the number of gene copies by $2N$, giving $4N/[k(k-1)]$. A mitochondrial sample naturally uses a haploid maternal effective population size; the common factor cancels in the ratios below.