Maximally Distant Trees and Principal Partition of a Linear Graph

A pair of trees of a graph is said to be maximally distant if the distance between these trees is maximum in the graph. Necessary and sufficient conditions for maximally distant trees are presented. Fundamental properties of the maximally distant trees provide us with the principal partition of a graph that is uniquely determined for the graph. Some useful properties of the principal partition, together with the application to the classification of the trees, are discussed.