  Recently, Abraham and Delmas constructed the distributions of super-critical
L\'evy trees truncated at a fixed height by connecting super-critical L\'evy
trees to (sub)critical L\'evy trees via a martingale transformation. A similar
relationship also holds for discrete Galton-Watson trees. In this work, using
the existing works on the convergence of contour functions of (sub)critical
trees, we prove that the contour functions of truncated super-critical
Galton-Watson trees converge weakly to the distributions constructed by Abraham
and Delmas.
