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Hospital readmissions have become one of the key measures of healthcare quality. Preventable readmissions have been identified as one of the primary targets for reducing costs and improving healthcare delivery. However, most data driven studies for understanding readmissions have produced black box classification and predictive models with moderate performance, which precludes them from being used effectively within the decision support systems in the hospitals. In this paper we present an application of structured sparsity-inducing norms for predicting readmission risk for patients based on their disease history and demographics. Most existing studies have focused on hospital utilization, test results, etc., to assign a readmission label to each episode of hospitalization. However, we focus on assigning a readmission risk label to a patient based on their disease history. Our emphasis is on interpreting the models to improve the understanding of the readmission problem. To achieve this, we exploit the domain induced hierarchical structure available for the disease codes which are the features for the classification algorithm. We use a tree based sparsity-inducing regularization strategy that explicitly uses the domain hierarchy. The resulting model not only outperforms standard regularization procedures but is also highly sparse and interpretable. We analyze the model and identify several significant factors that have an effect on readmission risk. Some of these factors conform to existing beliefs, e.g., impact of surgical complications and infections during hospital stay. Other factors, such as the impact of mental disorder and substance abuse on readmission, provide empirical evidence for several pre-existing but unverified hypotheses. The analysis also reveals previously undiscovered connections such as the influence of socioeconomic factors like lack of housing and malnutrition.