A transition from tree S to tree S
. Leaves 1, 2, 3 contain in-group species, leaf d* contains an auxiliary outgroup species. Leaf d* is connected to the root by the outgroup tube (shown in bold). All tubes acquire additional vertices during transition to S0 (right, shown in bold) to delimit time slices (here four slices are separated by dashed lines). Each slice thus contains one segment of the outgroup tube in S (left), which forms the outgroup tubes in S0 (each shown in bold). Any such segment, as well as the outgroup leaf-species, are denoted as d*. The root tube d0 is attached to the root, by analogy with the edge e0 in a gene tree G.