Comparing classification tree structures: a special case of comparing q-ary relations

Comparing q-ary relations on a set Q of elementary objects is one of the most fundamental problems of classification and combinatorial data analysis. In this paper the specific comparison task that involves classification tree structures (binary or not) is considered in this context. Two mathematical representations are proposed. One is defined in terms of a weighted binary relation; the second uses a 4-ary relation. The most classical approaches to tree comparison are discussed in the context of a set theoretic representation of these relations. Formal and combinatorial computing aspects of a construction method for a very general family of association coefficients between relations are presented. The main purpose of this article is to specify the components of this construction, based on a permutational procedure, when the structures to be compared are classification trees.