Figure 1.

A transition from tree S to tree S₀. 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.