Fig. 2

Left an example of a “true” history of a gene tree that evolves along the (tube-like) species tree (taken from [11]). The set of extant genes \(\mathbb {G}\) comprises \(a,b,c_1,c_2\) and d and \(\sigma\) maps each gene in \(\mathbb {G}\) to the species (capitals below the genes) \(A,B,C,D\in \mathbb {S}\). Upper right the observable gene tree \((T;t,\sigma )\) is shown. To derive \(\mathcal {S}(T;t,\sigma )\) we cannot use the triples \(\mathcal {R}_0(T)\), that is, we need to remove the transfer edges. To be more precise, if we would consider \(\mathcal {R}_0(T)\) we obtain the triples \(\mathsf {(ac_1|d)}\) and \(\mathsf {(c_2d|a)}\) which leads to the two contradicting species triples \(\mathsf {(AC|D)}\) and \(\mathsf {(CD|A)}\). Thus, we restrict \(\mathcal {R}_0\) to \(T_{\mathcal {\overline{E}}}\) and obtain \(\mathcal {R}_0(T_{\mathcal {\overline{E}}}) = \{\mathsf {(ac_1|d)}\}\). However, this triple alone would not provide enough information to obtain a species tree such that a valid reconciliation map \(\mu\) can be constructed. Hence, we take \(\mathcal {R}_1(T_{\mathcal {\overline{E}}})=\{\mathsf {(bc_2|d)}\}\) into account and obtain \(\mathcal {S}(T;t,\sigma ) = \{\mathsf {(AC|D)},\mathsf {(BC|D)}\}\). Lower right a least resolved species tree S (obtained with BUILD) that displays all triples in \(\mathcal {S}(T;t,\sigma )\) together with the reconciled gene tree \((T;t,\sigma )\) is shown. Although S does not display the triple \(\mathsf {(AB|C)}\) as in the true history, this tree S does not pretend a higher resolution than actually supported by \((T;t,\sigma )\). Clearly, as more gene trees (gene families) are available as more information about the resolution of the species tree can be provided