Figure 2

Let S =((1,2),3) (the tree in (a)) be the phylogenetic tree for the three species {1,2,3}. Let T=(1,2) be a gene tree. (a), (b) and (c) are the three possibilities for T after a random insertion of a leaf labelled 3. (a) is the only case leading to a tree without any NAD vertex. It reflects a history of the three gene copies without any duplication or loss; (b) and (c) each contains a NAD vertex, and can be explained by a duplication-loss history of minimum mutation cost of 4: 1 duplication and 3 losses.