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We consider the problem of computing the probability of a Boolean function, which generalizes the model counting problem. Given an OBDD for such a function, its probability can be computed in linear time in the size of the OBDD. In this paper we investigate the connection between treewidth and the size of the OBDD. Bounded treewidth has proven to be applicable to many graph problems, which are NP-hard in general but become tractable on graphs with bounded treewidth. However, it is less well understood how bounded treewidth can be used for the probability computation problem of a Boolean function. We introduce a new notion of treewidth of a Boolean function, called the expression treewidth, as the smallest treewidth of any DAG-expression representing the function. Our new notion of bounded treewidth includes some previously known tractable cases: all read-once Boolean functions, and all functions having a bounded treewidth of the primal graph or of the incidence graph also have a bounded expression treewidth. We show that bounded expression treewidth implies the existence of a polynomial size OBDD, and that bounded expression pathwidth implies the existence of a constant-width OBDD. We also show a converse of the latter result: constant-width OBDD imply bounded expression pathwidth. We then study the implications of these results to query compilation, where the Boolean function is the lineage of a fixed query on varying input databases. We give a syntactic characterizations of all UCQ≠ queries that admit a polynomial size OBDD, showing that these are precisely inversion-free queries with unrestricted use of ≠. It was previously known that inversion-free queries characterize precisely those UCQ queries that have a polynomial size OBDD, and that these also have a constant width OBDD: in contrast, inversion-free queries with ≠ have polynomial-width OBDD, thus using the full power of OBDD. Finally, we show that in the case of UCQ, the four classes studied in this paper collapse: bounded expression pathwidth, bounded expression treewidth, constant-width OBDD, and polynomial size OBDD.