Recent progress in per-pixel object class labeling of natural images can be attributed to the use of multiple types of image features and sound statistical learning approaches. Within the latter, Conditional Random Fields (CRF) are prominently used for their ability to represent interactions between random variables. Despite their popularity in computer vision, parameter learning for CRFs has remained difficult, popular approaches being cross-validation and piecewise training. In this work, we propose a simple yet expressive tree-structured CRF based on a recent hierarchical image segmentation method. Our model combines and weights multiple image features within a hierarchical representation and allows simple and efficient globally-optimal learning of 105 parameters. The tractability of our model allows us to pose and answer some of the open questions regarding parameter learning applying to CRF-based approaches. The key findings for learning CRF models are, from the obvious to the surprising, i) multiple image features always help, ii) the limiting dimension with respect to current models is the amount of training data, iii) piecewise training is competitive, iv) current methods for max-margin training fail for models with many parameters.