Building a recombining trinomial or multinomial tree under discrete-time generalized GARCH process is more difficult than one might have thought. Trinomial (or multinomial) trees are constructed to enable pricing of derivative securities under GARCH processes. The authors introduce important modifications to the method proposed by Ritchken and Trevor (1999). The algorithm is made more accurate by modifying the forward-building process and actually using interpolated variances only during the backward recursion process. This would yield a volatility pattern consistent with the true volatility process. The new algorithm, with its strong convergence properties, will compute the option prices very accurately. The modifications in this paper improve the model without making the algorithm less efficient.