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.