  In this work, we develop shape expansions of minimal matchbox manifolds
without holonomy, in terms of branched manifolds formed from their leaves. Our
approach is based on the method of coding the holonomy groups for the foliated
spaces, to define leafwise regions which are transversely stable and are
adapted to the foliation dynamics. Approximations are obtained by collapsing
appropriately chosen neighborhoods onto these regions along a "transverse
Cantor foliation". The existence of the "transverse Cantor foliation" allows us
to generalize standard techniques known for Euclidean and fibered cases to
arbitrary matchbox manifolds with Riemannian leaf geometry and without
holonomy. The transverse Cantor foliations used here are constructed by purely
intrinsic and topological means, as we do not assume that our matchbox
manifolds are embedded into a smooth foliated manifold, or a smooth manifold.
