Fig. 4

From the binary gene tree \((T;t,\sigma )\) (right) we obtain the species triples \(\mathcal {S}(T;t,\sigma ) = \{\mathsf {(AB|D)},\mathsf {(AC|D)}\}\). Shown are two (tube-like) species trees (left and middle) that display \(\mathcal {S}(T;t,\sigma )\). The respective reconciliation maps for T and S are given implicitly by drawing T within the species tree S. The left tree S is least resolved for \(\mathcal {S}(T;t,\sigma )\). Although there is even a unique reconciliation map from T to S, this map is not time-consistent. Thus, no time-consistent reconciliation between T and S exists. On the other hand, for T and the middle species tree \(S'\) (that is a refinement of S) there is a time-consistent reconciliation map. Fig. 2 provides an example that shows that also least-resolved species trees can have a time-consistent reconciliation map with gene trees