Fig. 1

Let v be the internal vertex associated with the subtree bipartition X|Y, where \(X = \{A\}\) and \(Y = \{B\}\) (note that these vertices are circled in the trees above). Subfigures A and B show two different trees on the same species set with Dollo-labelings for the same character. The state assignment at v only requires us to know the subtree bipartition associated with v (Corollary 1). Vertex v is assigned state 1 because there is a leaf in Y assigned state 1 and a leaf in \(S \setminus X \cup Y\) assigned state 1. Subfigures C and D show two different trees with Camin-Sokal-labelings for the same character. The state assignment at v only requires us to know the clade associated with v (Corollary 3). Vertex v is assigned state 0 because there is a leaf in clade \(X \cup Y\) assigned state 0. Lastly, subfigures E and F show two different trees with Fitch-labelings for the same character. In subfigure E, the assignment of state 0 or 1 to v results in a score of two or three, respectively (so 0 is better). In subfigure F, the assignment of state 0 or 1 to v results in a score of three or two, respectively (so 1 is better). Thus, for the Fitch criterion score, the state assignment at v depends on more than the bipartition induced by the edge incident to v