Total branch length of a neutral coalescent ($L={\sum }_{j=2}^{n}j{T}_j$)

In coalescent units, independent waiting times satisfy $T_j\sim \mathrm{Exp}(j(j-1)/2)$, and total branch length is $L={\sum }_{j=2}^{n}j{T}_j$. Therefore $EL=2{\sum }_{i=1}^{n-1}1/i$. The variables $j{T}_j/2$ are independent exponentials with rates $j-1$, so $W=L/2$ has the same distribution as the maximum of $n-1$ independent unit exponentials, with density $(n-1)e^{-w}(1-e^{-w}{)}^{n-2}$. The last identity follows either from its Laplace transform or from the independent spacings of exponential order statistics.