We define a class of properties on random plane trees, which we call subtree additive properties, inspired by the combinatorics of certain biologically-interesting properties in a plane tree model of RNA secondary structure. The class of subtree additive properties includes the Wiener index and path length (total ladder distance and total ladder contact distance, respectively, in the biological context). We then investigate the asymptotic distribution of these subtree additive properties on a random plane tree distributed according to a Gibbs distribution arising from the Nearest Neighbor Thermodynamic Model for RNA secondary structure. We show that for any property in the class considered, there is a constant that translates the uniformly weighted random variable to the Gibbs distribution weighted random variable (and we provide the constant). We also relate the asymptotic distribution of another class of properties, which we call simple subtree additive properties, to the asymptotic distribution of the path length, both in the uniformly weighted case. The primary proof techniques in this paper come from analytic combinatorics, and most of our results follow from relating the moments of known and unknown distributions and showing that this is sufficient for convergence.