  Fix an integer h>=1. In the universe of coloured trees of height at most h,
we prove that for any graph decision problem defined by an MSO formula with r
quantifiers, there exists a set of kernels, each of size bounded by an
elementary function of r and the number of colours. This yields two noteworthy
consequences. Consider any graph class G having a one-dimensional MSO
interpretation in the universe of coloured trees of height h (equivalently, G
is a class of shrub-depth h). First, class G admits an MSO model checking
algorithm whose runtime has an elementary dependence on the formula size.
Second, on G the expressive powers of FO and MSO coincide (which extends a 2012
result of Elberfeld, Grohe, and Tantau).
