= 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 $\binom{k}{2}$ pairs. Hence the mean <phylogenetic epoch> duration is
$$
\boxed{E_k^{\rm coal}=\frac{N}{\binom{k}{2}}=\frac{2N}{k(k-1)}}.
$$
Thus $E_2=N$, $E_3=N/3$ 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.
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