Fig. 10

An example of unrooted input trees \(G_I\) and \(S_I\) such that the species tree can be rooted at any point \(\text{ pt }(a,b,\rho )\) for \(\rho \in [1,4)\). The lower part of the figure shows separately solutions for \(\rho \in (1,3)\) and \(\rho \in (3,4)\). These two cases differ in the position of \(\Phi (x)\). For \(\rho =1\), node q coincides with x (and thus \(\Phi (q)\) coincides with \(\Phi (x)\)) and for \(\rho =3\), node r coincides with \(\Phi (x)\). Case \(\rho =4\) is not possible because our definitions do not allow rooting a phylogenetic tree in its leaf. Note that the species tree cannot be rooted at position \(\text{ pt }(a,b,\rho )\) for \(\rho \in (0,1)\)