On the graph of trees

We consider an ldquon-graph of treesrdquo whose nodes are the set of trees of fixed order n, and in which two nodes are adjacent if one tree can be derived from the other through a single application of a local edge transformation rule. We derive an exact formula for the length of the shortest path from any node to any ldquocanonicalrdquo node in the n-graph of trees. We use this result to derive upper and lower bounds on the diameter of the n-graph of trees. We then propose a coordinate system that is convenient for studying the structure of the n-graph of trees, and in which trees having the same degree sequence are projected onto a single point.