Distance measures between trees are useful for comparing trees in a systematic 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.
Sand, A., Holt, M. K., Johansen, J., Fagerberg, R., Brodal, G. S., Pedersen, C. N. S., & Mailund, T. (2013). Algorithms for Computing the Triplet and Quartet Distances for Binary and General Trees. Biology, 2(4), 1189-1209. https://doi.org/10.3390/biology2041189