Abstract: Considering the class Γ(n, m) of ordered trees with m leaves and n-m internal nodes, a set of generating functions are established for the following problems:(1) the total number nodes with degree r over Γ(n, m), (2) the total path length of nodes over Γ(n, m), and (3) the total number of nodes over Γ(n, m) on level k. Some particular counting fomulas are derived from them.