Fig. 2

When there are ambiguous states, we can restrict a character c and tree T to the subset of labels assigned non-ambiguous states (i.e., \(R = \{A, C, G, J, Out\}\)) and then compute the Dollo score in the usual fashion (see proof of Theorem 2). To create the restricted tree \(T|_R\), we first identify all edges incident to maximally-sized subtrees with all leaves assigned the ambiguous state (shown in red shorter dashes). After deleting these edges, we have a tree \(T'\) on R (shown with solid lines) and a collection P of subtrees (shown in grey, with edges as longer dashes) whose leaves are all assigned the ambiguous state. We then suppress vertices with out-degree 1 (shown in red) in \(T'\) to get \(T|_R\). Lastly, we apply conditions 1–3 to find the Dollo-labeling for the internal vertices of \(T|_R\); this gives us one loss on edge \(v \mapsto A\) (and also one gain on edge \(q \mapsto s\)). This procedure for constructing \(T|_R\) classifies vertices in T into three groups. The vertices in Group 1 (r, q, s, v) are assigned the same labels as in \(T|_R\). The vertices in Group 2 (x, z) are assigned the ambiguous state. The vertices in Group 3 (t, u, w, y) need to be assigned states so as not to increase the Dollo score. In this example, there are two possible ways to assign a state to vertex w; the approach described in the proof of Theorem 2 assigns state 0 to w