manner, and several different distance measures have been proposed. The triplet and quartet
distances, for rooted and unrooted trees, respectively, are defined as the number of subsets
of three or four leaves, respectively, where the topologies of the induced subtrees differ.
These distances can trivially be computed by explicitly enumerating all sets of three or four
leaves and testing if the topologies are different, but this leads to time complexities at least
of the order n3 or n4 just for enumerating the sets. The different topologies can be counted
implicitly, however, and in this paper, we review a series of algorithmic improvements that
have been used during the last decade to develop more efficient algorithms by exploiting
two different strategies for this; one based on dynamic programming and another based on
coloring leaves in one tree and updating a hierarchical decomposition of the other.