Figure 1.

Description of the two problems. (1) A gene tree (the "initial tree") for the gene family {c, b1, b2, a1, a2} is shown with small red nodes and single thin red edges. It is reconciled with the phylogeny of the three species A, B and C shown with large green nodes and hollow edges represented by a pair of parallel black lines. Duplication nodes of the reconciled gene tree are squared, while speciation nodes and leaves are dots. (2) The two neighbors of b1 on genome B and of a1 on genome A are inferred to be orthologous according to their lowest common ancestor in their respective gene trees (not shown). This is an argument for infering orthology between b1 and a1, which is in contradiction with the information provided by the initial tree: their lowest common ancestor is a duplication, and thus they are inferred to be paralogous. (3) A solution to the GOC problem, that is a gene tree of minimum RF distance with the initial tree verifying the constraint of b1 and a1 being orthologous. (4) A solution to the COC problem, that is a reconciled tree in which the clade {b1, b2, a1, a2} of d in the initial tree is rather rooted by a speciation node in the corrected tree. This is an example where the optimal solutions to the two problems differ.