Despite much work within the last decade on foundational properties of SPARQL - the standard query language for RDF data - rather little is known about the exact limits of tractability for this language. In particular, this is the case for SPARQL queries that contain the OPTIONAL-operator, even though it is one of the most intensively studied features of SPARQL. The aim of our work is to provide a more thorough picture of tractable classes of SPARQL queries. In general, SPARQL query evaluation is PSPACE-complete in combined complexity, and it remains PSPACE-hard already for queries containing only the OPTIONAL-operator. To amend this situation, research has focused on "well-designed SPARQL queries" and their recent generalization "weakly well-designed SPARQL queries". For these two fragments the evaluation problem is coNP-complete in the absence of projection and SigmaP2-complete otherwise. Moreover, they have been shown to contain most SPARQL queries asked in practical settings. In this paper, we study tractable classes of weakly well-designed queries in parameterized complexity considering the equivalent formulation as pattern trees. We give a complete characterization of the tractable classes in the case without projection. Moreover, we show a characterization of all tractable classes of simple well-designed pattern trees in the presence of projection.