Fig. 3

Top right: the gene tree \((T;t,\sigma )\) from Fig. 1 from which we obtain the species triplets \(\mathcal {R}(T;t,\sigma ) = \{AB|D,AC|D\}\). We start with the star tree \(S_1\) (top left) and obtain \(G((T; t, \sigma ), S_1, 1')\), which is shown right to \(S_1\). \(G((T; t, \sigma ), S_1, 1')\) has four vertices A, B, C, D and two edges. The edge labels indicate which of the conditions in Def. 9 yield the respective edge. In \(G((T; t, \sigma ), S_1, 1')\), there is only one non-trivial connected component which implies the good split that results in the tree \(S_2\) (lower left). There is only one cherry \(2'\) in \(S_2\) and the corresponding graph \(G((T; t, \sigma ), S_2, 2')\) is drawn right to \(S_2\). Again, the connected components give a good split that results in the binary tree \(S_3\). The tree \(S_3\) is precisely the species tree (planted root omitted) as shown in the middle of Fig. 1