  Enumerative studies of RNA secondary structures were initiated four decades
ago by Waterman and his coworkers. Since then, RNA secondary structures have
been explored according to many different structural characteristics, for
instance, helices, components and loops by Hofacker, Schuster and Stadler,
orders by Nebel, saturated structures by Clote, the $5^{\prime}$-$3^{\prime}$
end distance by Clote, Ponty and Steyaert, and the rainbow spectrum by Li and
Reidys. However, the majority of the contributions are asymptotic results, and
it is harder to derive explicit formulas. In this paper, we obtain exact
formulas counting RNA secondary structures with a given number of helices as
well as a given joint size distribution of helices and loops, while some
related asymptotic results due to Hofacker, Schuster and Stadler have been
known for about twenty years. Our approach is combinatorial, analyzing a recent
bijection between RNA secondary structures and plane trees discovered by the
first author and proposing a variation of Chen's bijective approach of counting
trees by forests of simple trees.
