and ${\textbf{Y}}_2$ can alternatively be represented graphically as an *augmented perfect phylogeny* ${\textbf{T}}$. This graphical representation of the data is exploited by our likelihood algorithm. The augmented perfect phylogeny representation is an extension of the *gene tree* or *perfect phylogeny* [@gus91; @gri94b; @pal19] representation to the heterochronous case. The standard perfect phylogeny definition leaves out the information carried by ${\textbf{Y}}_2$. To our knowledge, Gusfield’s approach has never been generalized to the heterochronous case. In the augmented perfect phylogeny ${\textbf{T}}=({\textbf{V}},{\textbf{E}})$, ${\textbf{V}}{}$ is the set of nodes of [**T**]{}, and ${\textbf{E}}{}$ is the set of weighted edges. We define [**T**]{} as follows:

1.  Each haplotype labels at least one leaf in ${\textbf{T}}$. If a haplotype is observed at $k$ different sampling times, then $k$ leaves in ${\textbf{T}}$ will be labeled by the same haplotype. The pair (haplotype label, sampling group) uniquely labels each leaf node.

2.  Each of the $z$ polymorphic sites labels exactly one edge. When multiple sites label the same edge, the order of the labels along the edge is arbitrary. Some external edges (edges subtending leaves) may not be labeled, indicating that they do not carry additional mutations to their parent node.

3.  For any pair (haplotype $h_{k}$, sampling 