Fig. 2

Type I and Type II superleaves. Trees T and t with edges in the backbone (defined to be the edges on paths between nodes in the common leaf set) colored green for shared, and blue for unique; the other edges are inside superleaves and are colored black. The deletion of the backbone edges in T creates two components that are called “superleaves”. One of the two superleaves is a Type I superleaf because it is attached to a shared (green) edge, and the other is a Type II superleaf because it is attached to a unique (blue) edge. The RF distance between t and \(T|_R\) is equal to 2, the number of blue edges. The Type I superleaf containing leaves r and s can be added to edge \(e_x\) in t, the shared edge incident to leaf x, without increasing the RF distance; adding it to any other edge in t will increase the RF distance by exactly 2. However, adding the Type II superleaf containing leaves u, v,  and q to any single edge in t creates exactly one new unique edge in each tree, and therefore increases the RF distance by exactly 2. More generally, for any pair of trees (one a gene tree and the other a reference tree), (1) any Type I superleaf can be added to the gene tree without increasing the RF distance, (2) any addition of a Type II superleaf to the gene tree will always increase the RF distance by at least 2, and (3) there is always at least one edge into which a Type II superleaf can be added that increases the RF distance by exactly 2