computing the quartet distance between general trees

2 CHAPTER 1. INTRODUCTIONhas is called the degree of the node. An unrooted tree does not contain internal nodes ofdegree two. A tree where internal nodes are allowed to be polytomies, that is, they canhave any degree equal to or greater than three, is called a general tree. General trees areoften used to represent partly resolved relationships where the complete topology is notknown and each species therefore cannot be represented by a distinct node. Sometimesbranches are assigned a length which adds the notion of time to the evolution.The true evolutionary relation among a set of EUs is rarely known. Multiple methodsfor determining the exact relationship, from biological data, are available. They do notnecessarily agree and might induce different trees. Some methods for inferring relationshipswill result in a large range of plausible tree reconstructions. Furthermore, multipledata sets, e.g. DNA sequences, describing a single species are often at hand. Thus, onemethod may yield a different solution for each data set used. Figure 1.1 is an illustrationof two alternative relationships inferred for the Panthera (big cats).PantheraClouded LeopardJaguarLeopardLionSnow LeopardTigerPantheraClouded LeopardTigerJaguarSnow LeopardLeopardLionFigure 1.1: Two alternative views of the relationship between the Panthera (bigcats). Note that one is a binary tree whereas the other includes a polytomy andis thus a general tree. Example from Davis et al. [9].This disagreement between trees introduces the need of some means of assessingtrees. One approach is to make a pairwise comparison of trees in an attempt of quantifyingthe differences or similarities.1.2 Measuring difference or similarityVarious methods for tree comparison have been defined and each measure has certainproperties and takes certain aspects of the trees into consideration. Some can only handlefully resolved trees while others are able to take branch lengths into account. Somemetrics consider topological properties only. An example of the latter is the nearest–neighbor interchange metric, proposed by Waterman and Smith [20], defined as thefewest number of nearest–neighbor interchanges required to convert one tree into another.The metric only works for binary trees and the problem of computing it has beenshown to be NP-complete (seeDasGupta et al. [8]). In this thesis focus will be on generaltrees. Here, an example is the Robinson–Foulds distance metric, proposed by Robinsonand Foulds [15], and also known as the symmetric difference metric. It is defined as the