Fig. 1

One iteration of the OCTAL algorithm. 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; all other edges are colored black. After rooting \(T|_{R}\) with respect to u, the edges in \(T|_{R}\) that could be identified by the algorithm for “placement” are indicated with an asterisk (*). Note that any path in \(T|_R\) from the root to a leaf will encounter a shared edge, since the edges incident with leaves are always shared. In this scenario, the edge e above the least common ancestor of leaves w and x is selected; this edge defines the same bipartition as edge \(e'\) in t. Hence, AddLeaf will insert leaf u into t by subdividing edge \(e'\), and making u adjacent to the newly added node