An infinite class of unsaturated rooted trees corresponding to designable RNA secondary structures

An RNA secondary structure is designable if there is an RNA sequence which can attain its maximum number of base pairs only by adopting that structure. The combinatorial RNA design problem, introduced by Hale\v{s} et al. in 2016, is to determine whether or not a given RNA secondary structure is designable. Hale\v{s} et al. identified certain classes of designable and non-designable secondary structures by reference to their corresponding rooted trees. We introduce an infinite class of rooted trees containing unpaired nucleotides at the greatest depth, and prove constructively that their corresponding secondary structures are designable. This complements previous results for the combinatorial RNA design problem.