Computing the Quartet Distance Between Trees of Arbitrary Degrees

Comparing trees with regard to their topology is in itself an interesting theoretical problem in computer science, and furthermore researchers working in the interdisciplinary field of computational biology need tools to compare phylogenetic trees, i.e. trees that describe the relation of species according to evolutionary history. Different methods and different information can result in different phylogenetic trees, and consequently there is a need to be able to compare such trees. Comparison of trees can be done by calculating the distance between them, and among the distance measures usable on trees are the quartet distance. A quartet is a set of four leaves in a tree, and the edges in the tree connecting the leaves imply the topology of the quartet. The quartet distance between two trees containing the same leaves is the number of quartets containing the same four leaves that have different topology in the two trees. Previous algorithms focus on calculating the quartet distance between binary trees. We explore different approaches for calculating the quartet distance between trees of arbitrary degrees. Each approach gives rise to one or two algorithms with varying running times and space consumptions. The running times are verified experimentally and a possibility for reducing the space consumption of the fastest algorithm is discussed. We have implemented the fastest algorithm in a tool, which is also presented, along with the feature of visualizing the similarity of trees using the quartet distance.