Comparison of Additive Trees Using Circular Orders

It has been postulated that existing species have been linked in the past in a way that can be described using an additive tree structure. Any such tree structure reflecting species relationships is associated with a matrix of distances between the species considered which is called a distance matrix or a tree metric matrix. A circular order of elements of X corresponds to a circular (clockwise) scanning of the subset X of vertices of a tree drawn on a plane. This paper describes an optimal algorithm using circular orders to compare the topology of two trees given by their distance matrices. This algorithm allows us to compute the Robinson and Foulds topologic distance between two trees. It employs circular order tree reconstruction to compute an ordered bipartition table of the tree edges for both given distance matrices. These bipartition tables are then compared to determine the Robinson and Foulds topologic distance, known to be an important criterion of tree similarity. The described algorithm has optimal time complexity, requiring O(n(2)) time when performed on two n x n distance matrices. It can be generalized to get another optimal algorithm, which enables the strict consensus tree of k unrooted trees, given their distance matrices, to be constructed in O(kn(2)) time.