  We study the Hausdorff and the box dimensions of closed invariant subsets of
the space of pointed trees, equipped with a pseudogroup action. This
pseudogroup dynamical system can be regarded as a generalization of a shift
space. We show that the Hausdorff dimension of this space is infinite, and the
union of closed invariant subsets with dense orbit and non-equal Hausdorff and
box dimensions is dense in this space.
  We apply our results to the problem of embedding laminations into
differentiable foliations of smooth manifolds. One of necessary conditions for
the existence of such an embedding is that the lamination must admit a
bi-Lipschitz embedding into a manifold. A suspension of the pseudogroup action
on the space of pointed graphs gives an example where this condition is not
satisfied, with Hausdorff dimension of the space of pointed trees being the
obstruction to the existence of such a bi-Lipschitz embedding.
