In this paper we propose a novel approach to identify and label the structural elements of furniture e.g. wardrobes, cabinets etc. Given a furniture item, the subdivision into its structural components like doors, drawers and shelves is difficult as the number of components and their spatial arrangements varies severely. Furthermore, structural elements are primarily distinguished by their function rather than by unique color or texture based appearance features. It is therefore difficult to classify them, even if their correct spatial extent were known. In our approach we jointly estimate the number of functional units, their spatial structure, and their corresponding labels by using reversible jump MCMC (rjMCMC), a method well suited for optimization on spaces of varying dimensions (the number of structural elements). Optionally, our system permits to invoke depth information e.g. from RGB-D cameras, which are already frequently mounted on mobile robot platforms. We show a considerable improvement over a baseline method even without using depth data, and an additional performance gain when depth input is enabled.