Algorithms for Computing the Triplet and Quartet Distances for Binary and General Trees

Abstract

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.

Identifying unstable taxa: Efficient implementation of triplet-based measures of stability, and comparison with Phyutility and RogueNaRok

“…It must be noted that the calculation of the triplet distance between two trees can in fact be done faster than O(t 3 ), with complexities as low as O(t 2 ) or even O(t ln t), with methods such as those reviewed by Sand et al (2013). However, those methods are based on simply calculating the number of triplets not shared by two trees, without identifiying them.…”

HIV-1 Full-Genome Phylogenetics of Generalized Epidemics in Sub-Saharan Africa: Impact of Missing Nucleotide Characters in Next-Generation Sequences

“…For each clade with at least four taxa, we calculated the proportion of unrooted, labeled subtrees of four taxa whose topologies differed between inferred and true clades (Quartet distance). 48 In addition, we evaluated the Kendall-Colijn distance on the same clades. 49 Tree distances typically scaled with clade size.…”

Log Transformation Improves Dating of Phylogenies

“…We run LSD, LF, and wLogDate on the IQTree from Volz and Frost 38 and use the BEAST trees from Dudas et al 5, which include 1000 sampled trees (BEAST-1000) and the Maximum clade credibility tree (BEAST-MCC). To root the IQTree, we search for the rooting position that minimizes the triplet distance [27] between the IQTree and the BEAST-MCC tree.…”