Taxonomies have been proposed numerous times in the literature in order to encode semantic relationships between classes. Such taxonomies have been used to improve classi fication results by increasing the statistical efficiency of learning, as similarities between classes can be used to increase the amount of relevant data during training. In this paper, we show how data-derived taxonomies may be used in a structured prediction framework, and compare the performance of learned and semantically constructed taxonomies. Structured prediction in this case is multi-class categorization with the assumption that categories are taxonomically related. We make three main contributions: (i) We prove the equivalence between tree-structured covariance matrices and taxonomies; (ii) We use this covariance representation to develop a highly computationally efficient optimization algorithm for structured prediction with taxonomies; (iii) We show that the taxonomies learned from data using the Hilbert-Schmidt Independence Criterion (HSIC) often perform better than imputed semantic taxonomies. Source code of this implementation, as well as machine readable learned taxonomies are available for download from https://github.com/blaschko/tree-structured-covariance.