Combinatorics of generalized orthogonal polynomials of type $R_{II}$

In 1995, Ismail and Masson introduced orthogonal polynomials of types $R_I$ and $R_{II}$, which are defined by specific three-term recurrence relations with additional conditions. Recently, Kim and Stanton found a combinatorial interpretation for the moments of orthogonal polynomials of type $R_I$ in the spirit of the combinatorial theory of orthogonal polynomials due to Flajolet and Viennot. In this paper, we push this combinatorial model further to orthogonal polynomials of type $R_{II}$. Moreover, we generalize orthogonal polynomials of type $R_{II}$ by relaxing some of their conditions. We then prove a master theorem, which generalizes combinatorial models for moments of various types of orthogonal polynomials: classical orthogonal polynomials, Laurent biorthogonal polynomials, and orthogonal polynomials of types $R_I$ and $R_{II}$.