= Solution
A <phylogenetic epoch> is a time interval between successive branching events during which the number of ancestral branches is constant. Read backward from the four sampled leaves: there are four branches until the first merger, then three until the next, then two until their common root. Write the random durations as $T_4,T_3,T_2$, with <expected values> $E_k=\mathbb E[T_k]$.
The original solution diagram below places time horizontally and labels each branch by its number of descendant leaves. Type A is the pectinate topology; type B is the balanced topology. The coloured time strips identify the three <phylogenetic epochs>; their drawn widths are schematic, not estimated times.
\Image[/past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2024/paper-1-epochs.png]
{title=Epochs and descendant counts for four sampled lineages}
{description=Time runs from the common root to the sampled leaves. The two panels distinguish the three-plus-one and two-plus-two root splits; numbers on branches count sampled descendants.}
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