The use of similarity and distance as a metric for the purpose of comparison and matching is pervasive in a wide variety of scientific fields, for example, bioinformatics and data mining. The study of the fundamental concepts as well as the design of computational algorithms related to similarity and distance is therefore of essential importance for many applications. Given the widespread use of similarity, little literature exists on the provision of a formal definition for similarity as a metric that is general enough in a wide variety of contexts. This book therefore provides a metric definition for similarity on the basis of generality. Mathematical properties for similarity and its interrelationship with distance are derived in accord with the metric definition. The work is extended to applications for normalized local similarity. Further topic deals with speeding up the computation for tree edit distance by exploiting certain structural regularities.