singleton child nodes, $$\label{leaf_all}
 P(V, \{E, E_{ch_1}, \ldots, E_{ch_k} \}, {\textbf{a}}_{i} \mid  {\textbf{g}},N_e, \mu) \propto (\mu l)^{|E|} e^{-\mu \mathcal{T}} \sum_{\textbf{R} \in \Pi({\textbf{E}}_{V})} \prod_{j=1}^k (\mu l_{R_j})^{|E_{ch_j}|},$$ where the first term on the r.h.s is defined exactly as the quantity on the r.h.s. of , while the second term corresponds to the probability of all possible different matchings between $R_{1},\ldots,R_{k}$, the first $k$ indexes such that ${\textbf{a}}_{i,R_{j}}=V$, and $|E_{ch_1}|,|E_{ch_2}|,\ldots,|E_{ch_k}|$, the $k$ numbers of mutations observed on the edges $E_{ch_1}, \ldots, E_{ch_k}$ leading to the child nodes of $V$. In this expression, $\Pi({\textbf{E}}_{V})$ is the set of all possible such matchings **R**.

Before defining $\Pi({\textbf{E}}_{V})$ more precisely, we make two observations. First, not all matchings are possible since not all leaf branches terminate at the same time (heterochronous sampling). Second, it is enough to consider the allocations that contribute to distinct likelihood values, *i.e.* allocations for which the underlying samples are “distinguishable" in the sense that they have a different number of mutations. We define $\Pi({\textbf{E}}_{V})$ as the set of all possible “distinct matchings of number of observed singleton mutations to singleton branches", that is, allocations which lead to a distinct likelihood values. To construct $ \Pi({\textbf{E}}_{V})$, we first partition the singleton edges 