the edge that subtends $V$. For the example in Figure \[perfect\_phylo\](B), ${\textbf{E}}_{V_{1}}=\{E_{1},E_{3},E_{4}\}$. Let ${\textbf{V}}^*$ be the set of all $V \in {\textbf{V}}$ such that $|V|>1$, then the likelihood function is defined as $$\begin{aligned}
\label{lik_2}
P(\textbf{Y}\mid {\textbf{g}}, N_e,\mu)&=\sum_{i=1}^{\#{\textbf{a}}} P(\textbf{Y}, {\textbf{a}}{}_i \mid {\textbf{g}}, N_e, \mu)\nonumber\\
&= \sum_{i=1}^{\#{\textbf{a}}} \prod_{V \in {\textbf{V}}^*} P(V, {\textbf{E}}_{V}, {\textbf{a}}{}_i  \mid {\textbf{g}},  N_e, \mu),\end{aligned}$$ where we recall that $\#{\textbf{a}}$ is the number of possible allocations, we have written $N_e=(N_e(t))_{t\geq 0}$, and $P(V, {\textbf{E}}_{V}, {\textbf{a}}{}_i \mid  {\textbf{g}}, N_e, \mu)$ is the probability of observing the mutations of the ${\textbf{E}}_{V}$ edges along the corresponding branches of ${\textbf{g}}{}$ defined by the mapping ${\textbf{a}}_i$ as follows. If $V$ has no singleton child nodes, then ${\textbf{E}}_{V}=\{E\}$ and $$\label{intern_mut}
P(V, \{E\}, {\textbf{a}}_{i} \mid  {\textbf{g}}, N_e, \mu)  \propto (\mu l)^{|E|} e^{-\mu \mathcal{T}},$$ where $l$ is the length of the branch in ${\textbf{g}}$ that subtends ${\textbf{g}}_{j}$, $j$ is the largest index such that ${\textbf{a}}_{i,j}=V$, and $\mathcal{T}$ denotes the length of the subtree in ${\textbf{g}}$ to which $V$ is mapped in ${\textbf{a}}_{i}$ (as described in Subsection \[sec:alloc\]). For example, considering $V_2$ in Figure \[sub\_all\], we have $\mathcal{T}_2= 2 t_n +(t_{n-2}-t_n)$ and $l=(t_{n-2}-t_n)$ is the length of the branch connecting vintage $1$ to vintage $3$.

If node $V$ has 