Fig. 1

Left an example of a “true” history of a gene tree that evolves along the (tube-like) species tree. The set of extant genes \(\mathbb {G}\) comprises a,a′,b,b′,c,c′,c″ and e and \(\sigma\) maps each gene in \(\mathbb {G}\) to the species (capitals below the genes) \(A,B,C,E\in \sigma (\mathbb {G})\). For simplicity all speciation events followed by a loss along the path from v to \(a'\) in T are omitted. Left the observable gene tree \((T;t,\sigma )\) is shown. Since there is a true scenario which explains \((T;t,\sigma )\), the gene tree is biologically feasible. In particular, \((T;t,\sigma )\) satisfies (O1), (O2) and (O3)