A strict upper bound for the partition distance and the cluster distance of phylogenetic trees for each fixed pair of topological trees

For each given pair of (rooted or unrooted) topological trees with the same number of leaves a strict upper bound is shown for the tree partition distance (also called symmetric difference metric and Robinson-Foulds distance)—in case of unrooted trees—and for the cluster distance (also called Robinson-Foulds distance)—in case of rooted trees—of corresponding phylogenetic trees. In particular, it is shown that there exist assignments of labels (e.g., species) to the leaves of both topological tree where each label is assigned to exactly one leaf in each tree such that: i) in the unrooted case, the tree partition distance between the corresponding phylogenetic trees equals the number of internal edges in both trees minus the number of nodes with degree 2 in both trees, ii) in the rooted case, the cluster distance between any two corresponding phylogenetic trees equals the number of internal edges in both trees minus the number of nodes with degree 2 in both trees, and iii) the values in (i) and (ii) are also the maximum values with respect to all possible assignments. The shown strict worst case bounds are needed as normalization factor to compute a normalized version of the respective tree partition metrics.