Fig. 5

Incongruence of gene and species quartets implies the existence of an ancient duplication. Consider four pairwise distinct species A, B, C, and D whose species tree is given on the l.h.s., and let four genes a, b, c, and d be chosen such that \(\sigma (a)=A\), \(\sigma (b)=B\), \(\sigma (c)=C\), and \(\sigma (d)=D\). The two speciation events separating A from B and C from D are indicated by \(\bigcirc\). The root of this tree is indicated by \(\blacksquare\). Of the three possible gene quartets, one is congruent with the species tree. The other two are incongruent. In each of these, Eq. (2) implies that the two interior vertices in these quartets cannot be mapped to the species tree below the root. The root of the gene tree must thus be mapped above the root of the species tree