  Transductions are a general formalism for expressing transformations of
graphs (and more generally, of relational structures) in logic. We prove that a
graph class $\mathscr{C}$ can be $\mathsf{FO}$-transduced from a class of
bounded-height trees (that is, has bounded shrubdepth) if, and only if, from
$\mathscr{C}$ one cannot $\mathsf{FO}$-transduce the class of all paths. This
establishes one of the three remaining open questions posed by Blumensath and
Courcelle about the $\mathsf{MSO}$-transduction quasi-order, even in the
stronger form that concerns $\mathsf{FO}$-transductions instead of
$\mathsf{MSO}$-transductions.
  The backbone of our proof is a graph-theoretic statement that says the
following: If a graph $G$ excludes a path, the bipartite complement of a path,
and a half-graph as semi-induced subgraphs, then the vertex set of $G$ can be
partitioned into a bounded number of parts so that every part induces a cograph
of bounded height, and every pair of parts semi-induce a bi-cograph of bounded
height. This statement may be of independent interest; for instance, it implies
that the graphs in question form a class that is linearly $\chi$-bounded.
