= Solution
Count the number of descendant leaves on each branch during each <phylogenetic epoch>. From the tips backward, both topologies have branch-size lists $(1,1,1,1)$ during $T_4$, then $(2,1,1)$ during $T_3$. During $T_2$, type A has $(3,1)$ and type B has $(2,2)$.
Let $L_j$ be the total length of branches carrying $j$ descendant leaves. Adding the relevant <phylogenetic epoch> durations gives
$$
\boxed{\begin{array}{c|ccc}
&\mathbb E[L_1]&\mathbb E[L_2]&\mathbb E[L_3]\\\hline
A&4E_4+2E_3+E_2&E_3&E_2\\
B&4E_4+2E_3&E_3+2E_2&0
\end{array}}.
$$
For example, type A has one external branch and one three-descendant branch throughout $T_2$, while type B has two two-descendant branches. In either case the total expected branch length is $4E_4+3E_3+2E_2$, a useful accounting check. The stem above the common root is excluded because <mutations> there would be shared by all four samples.
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