Tree structured data often appear in bioinformatics. For example, glycans, RNA secondary structures and phylogenetic trees usually have tree structures. Comparison of trees is one of fundamental tasks in analysis of these data. Various distance measures have been proposed and utilized for comparison of trees, among which extensive studies have been done on tree edit distance. In this paper, we review key results and our recent results on the tree edit distance problem and related problems. In particular, we review polynomial time exact algorithms and more efficient approximation algorithms for the edit distance problem for ordered trees, and approximation algorithms for the largest common sub-tree problem for unordered trees. We also review applications of tree edit distance and its variants to bioinformatics with focusing on comparison of glycan structures.

It is known that the problem of finding a largest common subgraph is NP-hard for general graphs even if the number of input graphs is two. It is also known that the problem can be solved in polynomial time if the input is restricted to two trees. In this paper, a randomized parallel (an RNC) algorithm for finding a largest common subtree of two trees is presented. The dynamic tree contraction technique and the RNC minimum weight perfect matching algorithm are used to obtain the RNC algorithm. Moreover, an efficient NC algorithm is presented in the case where input trees are of bounded vertex degree. It works in O(log(n1)log(n2)) time using O(n1n2) processors on a CREW PRAM, where n1 and n2 denote the numbers of vertices of input trees. It is also proved that the problem is NP-hard if the number of input trees is more than two. The three dimensional matching problem, a well known NP-complete problem, is reduced to the problem of finding a largest common subtree of three trees.