Figure 1.

Consistency of labels at the backbone boundaries. At first (left) an initial backbone exists (thick branches), all inner nodes are labelled as inside (I) and each tip node has a cluster id. After completion of the post-order traversal (right), each inner node has been relabelled accordingly, if required. Here, cluster 2 is monophyletic, hence the cluster id was inherited propagated back to the initial backbone node. This produced a branch (edge) with an inside and an outside node; therefore the outside(O) node is relabelled (arrow) as boundary(B) node.