||RNA (Ribonucleic acid) is an important molecule in the cell, which participates in a lot of basic biological functions including regulation of DNA replication, synthesis of protein, and construction of RNA virus. The study of RNA has become a popular research topic. Unlike the helix structure of DNA, various structures can be formed by an RNA primary structure. Since most functions of RNA depend on their secondary structures, the enumeration of RNA secondary structures is very important.
In this thesis, we study combinatorial properties of RNA secondary structures. In the first part, we simplify the element definitions of RNA secondary structures so that they can be well-classified. We also propose skeleton trees to concisely represent the classes. By generating functions, we enumerate various classes of RNA secondary structures and point out incorrect results in literatures. In addition, some classes of RNA secondary structures with pseudoknots are enumerated.
In the second part, we concentrate on saturated RNA secondary structures. A saturated secondary structure is an RNA structure in which no additional basepair can be added. In the Nussinov-Jacobson free energy model, a secondary structure with maximum number of basepairs will have minimum free energy, and thus is most stable so that it is called an optimal structure. A t-saturated secondary structure contains fewer basepairs than an optimal one. In this thesis, we study properties of saturated secondary structures, enumerate near-optimal t-saturated secondary structures, and extended study some related topics.