Fig. 2

Top left: the gene tree \((T;t,\sigma )\) from Fig. 1 from which we obtain the species triplets \(\mathcal {R}(T;t,\sigma ) = \{AB|D,AC|D\}\). Moreover, the sequence of species trees \(S_1, S_2\) and \(S_3\) is obtained by stepwise application of good split refinements. The species tree \(S_4\) is an example of a split refinement of \(S_2\) that is not good. The corresponding graphs \(A({T},{S})\) are drawn right to the respective species tree S. For clarity, we have omitted to draw all vertices of \(A({T},{S})\) that have degree 0. Moreover, in the respective species trees the planted roots are omitted. See text for further discussion