Multivariate count data is ubiquitous in many areas of research including the physical, biological, and social sciences. These data are traditionally modeled with the Dirichlet Multinomial distribution (DM). A new, more flexible Dirichlet-Tree Multinomial (DTM) model is gaining in popularity. Here, we consider Bayesian DTM regression models. Our contribution is to introduce a novel shared shrinkage prior for use on the regression coefficients. The proposed prior enables branches in the Dirichlet trees to borrow information from one another, which encourages similar levels of shrinkage on covariates throughout the model. We focus on modeling multivariate count data in social and community settings like educational outcomes, crime rates, and voting data. In these setting, the interpretation of the regression coefficients is of particular interest. With that in mind, we pay special attention to both the interpretation of model parameters and the process of selecting a tree structure. A simulation study demonstrates the benefits of our proposed shared shrinkage prior against existing alternatives. We show the usefulness of the proposed prior in the analyses of two real datasets. The first dataset examines connections between household conditions and post-graduation intentions for high school seniors in Iowa's public school districts. The second dataset looks at the relationship between community characteristics and instances of different categories of crimes as reported by the FBI. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]