Transitivity of trees.

For a graph G = (V , E) , a partition π = (V 1 , V 2 , ... , V k) of the vertex set V is a transitive partition if V i dominates V j whenever i < j. The transitivity  Tr (G) of a graph G is the maximum order of a transitive partition of V. For any positive integer k , we characterize the smallest tree with transitivity k and obtain an algorithm to determine the transitivity of any tree of finite order. [ABSTRACT FROM AUTHOR]