Efficient Algorithms for Descendent Subtrees Comparison of Phylogenetic Trees with Applications to Co-evolutionary Classifications in Bacterial Genome

Abstract

Abstract. A phylogenetic tree is a rooted tree with unbounded degree such that each leaf node is uniquely labelled from 1 to n. The descendent subtree of of a phylogenetic tree T is the subtree composed by all edges and nodes of T descending from a vertex. Given a set of phylogenetic trees, we present linear time algorithms for finding all leaf-agree descendent subtrees as well as all isomorphic descendent subtrees. The normalized cluster distance, d(A, B), of two sets is defined by d(A, B) = ∆(A, B)/(|A| + |B|), where ∆(A, B)denotes the symmetric set difference of two sets. We show that computing all pairs normalized cluster distances between descendent subtrees of two phylogenetic trees can be done in O(n 2 ) time. Since the total size of the outputs will be Θ(n 2 ), the algorithm is thus computationally optimal. A nearest subtree of a subset of leaves is such a descendent subtree that has the smallest normalized cluster distance to these leaves. Here we show that finding nearest subtrees for a collection of pairwise disjointed subsets of leaves can be done in O(n) time. Several applications of these algorithms in areas of bioinformatics is considered. Among them, we discuss the 2CS (Two component systems) functional analysis and classifications on bacterial genome.

Common intervals of trees

“…A similar problem is the leaf-agree subtree problem [11], also mentioned as the common subtree problem in [15]. Starting from k rooted trees of size n with only leaves carrying labels, the problem is to find all k-tuples of induced subtrees with the same set of leaf labels (leaf-agree subtrees).…”

“…The major difference between common intervals and leaf-agree subtrees is that the latter do not consider interior nodes. The number of non-trivial leaf-agree subtrees is bounded by the number of interior nodes, and they can be found in O(kn) time [11].…”