computing the quartet distance between general trees

Chapter 5Calculating leaf set sizesCalculating the size of every subtree leaf set in a given tree, that is, the number of leavescontained in the subtree, is an essential part of two of the algorithms studied in thisthesis, namely the cubic time algorithm of Chap. 6 and, most important, the sub-cubicalgorithm of Chap. 7, that is the central topic of the thesis. So is the calculation of sharedleaf set sizes between two trees. A shared leaf set is the intersection of a pair of subtreeleaf sets. It turns out that the former is needed under fast calculation of the latter.Christiansen and Randers [5] study these calculations for different types of trees andhave been my primary inspiration. Here I will focus solely on outlining the approach andaspects important to this thesis.5.1 Subtree leaf set sizesThe calculation of all subtree leaf set sizes can be done in time O(n). Consider somegeneral unrooted tree, T like the one in Fig. 5.1(a). If T is rooted in one of its internalnodes, e.g. r , one can consider directed edges as either pointing away from r or towardsr , like e and e 2 respectively, see Fig. 5.1(b). Each directed edge represents a subtree. Thesubtree F , represented by e, does not contain r and the subtree ¯F , represented by theopposite edge, ē, does does contain r .The leaf set size of every subtree that, like F , does not contain r , can be calculatedrecursively using the following recipe. Make a depth-first traversal of all nodes, v ∈ T ,starting at r . For each node v we look at the subtree defined by the edge entering v andpointing away from r . If v is a leaf node, the leaf set size is 1. If v is an internal node,the leaf set size is the sum of all leaf set sizes of the subtrees pointing downwards from v.Half of the leaf set sizes have been calculated and this calculation takes O(n) time, sincea tree contains a linear number of nodes.25