Figure 6

Running example for the definitions and proof of Theorem 2. A running example for the definitions and proof of Theorem 2. Arrows are marked by numbers 1 to 6, demonstrating the steps of the proof. Each step is explained below: (1) Given an instance I = (T₁,...,T _n ), let H be the strict consensus tree of I. An internal node of H and its four children are shown. (2) Let S be a solution for I, having the optimal deep coalescence cost. (3) Cut trees via H and h, obtaining $\mathrm{C}\mathrm{u}{\text{t}}_{H,h}\left(I\right)$ and $\mathrm{C}\mathrm{u}{\text{t}}_{H,h}\left(S\right)=S^{\prime }.$ Let A, B, C, D be the leaf sets of each subtree. (4) We assume by contradiction that S' is not a solution for $\mathrm{C}\mathrm{u}{\text{t}}_{H,h}\left(I\right)$, and so we let R' be a solution for $\mathrm{C}\mathrm{u}{\text{t}}_{H,h}\left(I\right)\mathsf{\text{.}}$ (5 and 6) Modify S to obtain R, by replacing S' with R'.