  We first observe that the relations of the canonical generating isometries of
the Cuntz algebra ${\cal O}_N$ are naturally related to the $N$-colored Catalan
numbers. For a directed graph $G$, we generalize the Catalan numbers by using
the canonical generating partial isometries of the Cuntz-Krieger algebra ${\cal
O}_{A^G}$ for the transition matrix $A^G$ of $G$. The generalized Catalan
numbers $c_n^G, n=0,1,2,...$ enumerate the number of Dyck paths and oriented
rooted trees for the graph $G$. Its generating functions will be studied.
