  We show joint convergence of the Lukasiewicz path and height process for
slightly supercritical Galton-Watson forests. This shows that the height
processes for supercritical continuous state branching processes as constructed
by Lambert (2002) are the limit under rescaling of their discrete counterparts.
Unlike for (sub-)critical Galton-Watson forests, the height process does not
encode the entire metric structure of a supercritical Galton-Watson forest. We
demonstrate that this result is nonetheless useful, by applying it to the
configuration model with an i.i.d. power-law degree sequence in the critical
window, of which we obtain the metric space scaling limit in the product
Gromov-Hausdorff-Prokhorov topology, which is of independent interest.
