group), the labels of the edges along the unique path from the root to the leaf $h_{k}$ specify all the sites where $h_{k}$ has the mutant type.

Figure \[perfect\_phylo\](B) plots [**T**]{} corresponding to ${\textbf{Y}}_1$ and ${\textbf{Y}}_2$ displayed in Figure \[perfect\_phylo\](A). Observe that [**T**]{} includes sampling information in the leaf labels. In the example, $h_C$ labels two leaves because it is observed at times $s_1$ and $s_2$. The corresponding edges $E_3$ and $E_4$ are unlabeled, *i.e.*, no mutations are allocated to those edges because the underlying nodes carry identical sequences (same haplotype). We “augment" Gusfield’s perfect phylogeny because the sampling information is crucial in the likelihood calculation. [**T**]{} implicitly carries some quantitative information that can be quickly summarized. We denote the number of observed sequences subtended by an internal node $V$ by $|V|$. If $V$ is a leaf node, $|V|$ denotes the frequency of the haplotype $h$ observed at the corresponding sampling time $s$. Similarly, we denote the number of mutation labels assigned to an edge $E$ by $|E|$. If no mutations are assigned to $E$, then $|E|=0$. For parsimony, the edge that connects node $V_i$ to its parent node is denoted by $E_i$. See Figure \[perfect\_phylo\](C) for an example.

[@gus91] 