On the height of trees

In this note we shall deal with the enumeration of labelled trees of given order and given height over a selected point. An undirected graph is called a tree if it is connected and contains no cycle. If we select any two vertices P and Q of a tree T, there is evidently a uniquely determined path in T leading from P to Q. We shall call the length of this path (i.e. the number of edges in the path) the distance of P and Q in T and denote it by dT(P, Q). If a vertex P is distinguished as the root of T, we define the height of T over P as the length of the longest path in T starting from P; thus if hP(T) denotes the height of T over the root P, we have