Fig. 1

Left: A “true” evolutionary scenario for a gene tree with leaf set \(\mathbb {G}\) evolving along the tube-like species trees is shown. The symbol “x” denotes losses. All speciations along the path from the root \(\rho _T\) to the leaf a are followed by losses and we omit drawing them. Middle: The observable gene tree is shown in the upper-left. The orthology graph \(G = (\mathbb {G},E)\) (edges are placed between genes x, y for which \(t({\text {lca}}(x,y)) = \bullet\)) is drawn in the lower part. This graph is a cograph and the corresponding non-binary gene tree T on \(\mathbb {G}\) that can be constructed from such data is given in the upper-right part (cf. [3, 4, 14] for further details). Right: Shown is species trees S on \(\mathbb {S}=\sigma (\mathbb {G})\) with reconciled gene tree T. The reconciliation map \(\mu\) for T and S is given implicitly by drawing the gene tree T within S. Note, this reconciliation is not consistent with DTL-scenarios [20, 24]. A DTL-scenario would require that the duplication vertex and the leaf a are incomparable in S. Note, a non-binary duplication or HGT vertex v can always be “binary resolved” such that the newly created vertices are placed on the same edge \(\mu (v)\) as v. However, there are cases that show that non-binary speciation vertices cannot be “binary resolved”. For instance, for the non-binary gene tree T there is no way to resolve its root without violating the conditions of a reconciliation map (cf. [15, Fig. 3]). Yet, such cases strongly imply that the speciation event must have been followed by (several) duplication/HGT events that are not observable due to losses