In this paper, we introduce Multi-Outcome Regression Estimation with Tree-structured Shrinkage (MORETreeS), a class of Bayesian regression models for use in estimating the effect of a common exposure on a high-dimensional outcome whose variables are structured as a tree. MORETreeS employs a general class of tree-structured priors over the coefficients that leads to shrinkage of the parameters for related outcomes towards one another, with the potential for fusion. Here, we focus on binary data. In this context, the resulting posterior inference reduces the effective number of outcomes by automatically collapsing/combining related outcomes when effects appear homogeneous and/or when there are very few events. Distinct coefficients are estimated when there is sufficient evidence of exposure effect heterogeneity across the individual outcomes. As a motivating example, we consider the effect of average daily temperature on hospitalizations for thousands of different causes that are classified according to the hierarchical International Classification of Diseases 9. We employ data on hospitalizations among Medicare recipients in five major US cities from 1999 to 2010.