 Solution

In the infinite sites mutation model, mutations occur as independent Poisson processes at rate $\theta /2$ per lineage per unit of the coalescent time used in part (a). Conditional on the genealogy, superposition over branches gives a Poisson distribution with mean $\theta L/2$. Since this depends only on $L$, conditioning on total length alone givesEach mutation occurs at a new site and below the common ancestor, so it contributes exactly one segregating site. The law of iterated expectation therefore gives, with $a_{n-1}={\sum }_{i=1}^{n-1}1/i$,Here $\theta$ is fixed when taking the expectation. If it is also random under a proper prior with finite mean, the unconditional expectation is $a_{n-1}E\theta$.