We study plane trees as a model for RNA secondary structure, assigning energy to each tree based on the Nearest Neighbor Thermodynamic Model, and defining a corresponding Gibbs distribution on the trees. Through a bijection between plane trees and 2-Motzkin paths, we design a Markov chain converging to the Gibbs distribution, and establish fast mixing time results by estimating the spectral gap of the chain. The spectral gap estimate is established through a series of decompositions of the chain and also by building on known mixing time results for other chains on Dyck paths. In addition to the mathematical aspects of the result, the resulting algorithm can be used as a tool for exploring the branching structure of RNA and its dependence on energy model parameters. The pseudocode implementing the Markov chain is provided in an appendix.