Stochastic Lipschitz bandit algorithms balance exploration and exploitation, and have been used for a variety of important task domains. In this paper, we present a framework for Lipschitz bandit methods that adaptively learns partitions of context-and arm-space. Due to this flexibility, the algorithm is able to efficiently optimize rewards and minimize regret, by focusing on the portions of the space that are most relevant. In our analysis, we link tree-based methods to Gaussian processes. In light of our analysis, we design a novel hierarchical Bayesian model for Lipschitz bandit problems. Our experiments show that our algorithms can achieve state-of-the-art performance in challenging real-world tasks such as neural network hyperparameter tuning.