The combinatorial RNA design problem for binary trees

The nucleotides adenine, uracil, guanine, and cytosine are the building blocks of ribonucleic acid (RNA). Certain nucleotides can pair, creating folds in the RNA sequence known as the secondary structure. The stability of the secondary structure increases with the number of pairings, but there are typically many foldings that achieve the maximum number of pairings. The combinatorial RNA design problem is to find a design for a target secondary structure (a sequence which can achieve its maximum number of pairings only by folding into this structure), or else to show that no design exists for this structure. A design is known for a class of secondary structures in which all nucleotides are paired, but a structure in which even one nucleotide is unpaired need not admit a design. We prove constructively that there is an infinite class of secondary structures containing unpaired nucleotides and admitting a design.