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,...,Tn), 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 <a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S10/S12/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S10/S12/mathml/M93">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S10/S12/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S10/S12/mathml/M94">View MathML</a> Let A, B, C, D be the leaf sets of each subtree. (4) We assume by contradiction that S' is not a solution for <a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S10/S12/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S10/S12/mathml/M95">View MathML</a>, and so we let R' be a solution for <a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S10/S12/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S10/S12/mathml/M96">View MathML</a> (5 and 6) Modify S to obtain R, by replacing S' with R'.

Lin et al. BMC Bioinformatics 2012 13(Suppl 10):S12   doi:10.1186/1471-2105-13-S10-S12