Many properties of pairs of trees of a graph are related to the Hamming distance between them. This is important for several graph-theoretical concepts that have featured in hybrid graph theory. Here the notions of perfect pairs and superperfect pairs of trees have played a part. We define and characterize these notions in this chapter and describe necessary conditions for the unique solvability of affine networks in terms of trees and pair of trees.

The small number of theorems and propositions collected together in the opening paragraphs of Chapter 3 will again be frequently referred to here. Familiarity with the basic concepts of graphs such as circuit and cutset are presumed in this chapter. A maximal circuit-less subset of a graph G is called a tree of G while a maximal cutset-less subset of edges is called a cotree. These terms (circuit, cutset, tree and cotree) will be used here to mean a subset of the edges of a graph. Let F be a subset of E. Then the rank of F, denoted by rank (F), is the cardinality of a maximum circuit-less subset of F and the corank of F, denoted by corank (F), is the cardinality of a maximum cutsetless subset of F. The complement of F is the set difference E\F, denoted by F*. By |F| we denote the number of elements in (that is, the cardinality of) the subset F.

Diameter of a tree

Given a graph G, each its tree t can be classified according to the non-negative integer rank(f).