We train random and boosted forests, two machine learning architectures based on regression trees, to emulate a physics-based parameterization of atmospheric gravity wave momentum transport. We compare the forests to a neural network benchmark, evaluating both offline errors and online performance when coupled to an atmospheric model under the present day climate and in 800 and 1200 ppm CO2 global warming scenarios. Offline, the boosted forest exhibits similar skill to the neural network, while the random forest scores significantly lower. Both forest models couple stably to the atmospheric model, and control climate integrations with the boosted forest exhibit lower biases than those with the neural network. Integrations with all three data-driven emulators successfully capture the Quasi-Biennial Oscillation (QBO) and sudden stratospheric warmings, key modes of stratospheric variability, with the boosted forest more accurate than the random forest in replicating their statistics across our range of carbon dioxide perturbations. The boosted forest and neural network capture the sign of the QBO period response to increased CO2, though both struggle with the magnitude of this response under the more extreme 1200 ppm scenario. To investigate the connection between performance in the control climate and the ability to generalize, we use techniques from interpretable machine learning to understand how the data-driven methods use physical information. We leverage this understanding to develop a retraining procedure that improves the coupled performance of the boosted forest in the control climate and under the 800 ppm CO2 scenario.