Numerical similarity and dissimilarity measures between two trees

Quantifying the measure of similarity between two trees is a problem of intrinsic importance in the study of algorithms and data structures and has applications in computational molecular biology, structural/syntactic pattern recognition and in data management. In this paper we define and formulate an abstract measure of comparison, Ω(T-1, T-2), between two trees T-1 and T-2 presented in terms of a set of elementary intersymbol measures ω(., .) and two abstract operators + and x. By appropriately choosing the concrete values for these two operators and for ω(., .), this measure can be used to define various quantities including 1) the edit distance between two trees, 2) the size of their largest common subtree, 3) Prob(T-2, T-1), the probability of receiving T-2 given that T-1 was transmitted across a channel causing independent substitution and deletion errors, and 4) the a posteriori probability of T-1 being the transmitted tree given that T is the received tree containing independent substitution, insertion and deletion errors. The recursive properties of Ω(T-1, T-2) have been derived and a single generic iterative dynamic programming scheme to compute all the above quantities has been developed. The time and space complexities of the algorithm have been analyzed and the implications of our results in both theoretical and applied fields has been discussed.