It has been postulated that existing species have been linked in the past i
n a way that can be described using an additive tree structure. Any such tr
ee structure reflecting species relationships is associated with a matrix o
f distances between the species considered which is called a distance matri
x 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 draw
n on a plane. This paper describes an optimal algorithm using circular orde
rs to compare the topology of two trees given by their distance matrices, T
his algorithm allows us to compute the Robinson and Foulds topologic distan
ce between two trees, It employs circular order tree reconstruction to comp
ute an ordered bipartition table of the tree edges for both given distance
matrices, These bipartition tables are then compared to determine the Robin
son and Foulds topologic distance, known to be an important criterion of tr
ee similarity. The described algorithm has optimal time complexity, requiri
ng O (n(2)) time when performed on two n x It distance matrices. It can be
generalized to get another optimal algorithm, which enables the strict cons
ensus tree of k unrooted trees, given their distance matrices, to be constr
ucted in O (kn(2)) time.