  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.
