In this paper, we are concerned with identifying a subclass of tree adjoining grammars (TAGs) that is suitable for the application to modeling and predicting RNA secondary structures. The goal of this paper is twofold: For the purpose of applying to the RNA secondary structure prediction problem, we first introduce a special subclass of TAGs and develop a fast parsing algorithm for the subclass, together with some of its language theoretic characterizations. Then, based on the algorithm, we develop a prediction system and demonstrate the effectiveness of the system by presenting some experimental results obtained from biological data, where free energy evaluation selection for parse trees is incorporated into the algorithm.