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Hedonic games are a well-studied model of coalition formation, in which selfish agents are partitioned into disjoint sets, and agents care about the make-up of the coalition they end up in. The computational problem of finding a stable outcome tends to be computationally intractable, even after severely restricting the types of preferences that agents are allowed to report. We investigate a structural way of achieving tractability, by requiring that agents' preferences interact in a well-behaved manner. Precisely, we show that stable outcomes can be found in linear time for hedonic games that satisfy a notion of bounded treewidth and bounded degree.