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We prove, in the universe of trees of bounded height, that for any MSO formula with m variables there exists a set of kernels such that the size of each of these kernels can be bounded by an elementary function of m. This yields a faster MSO model checking algorithm for trees of bounded height than the one for general trees. From that we obtain, by means of interpretation, corresponding results for the classes of graphs of bounded tree-depth (MSO 2 ) and shrub-depth (MSO 1 ), and thus we givedoi:10.4230/lipics.fsttcs.2012.112 dblp:conf/fsttcs/GajarskyH12 fatcat:wit3in442fe4pn23tikkr6d54y