  In this paper, we prove that the self-dual morphological hierarchical
structure computed on a n-D gray-level wellcomposed image u by the algorithm of
G{\'e}raud et al. [1] is exactly the mathematical structure defined to be the
tree of shape of u in Najman et al [2]. We recall that this algorithm is in
quasi-linear time and thus considered to be optimal. The tree of shapes leads
to many applications in mathematical morphology and in image processing like
grain filtering, shapings, image segmentation, and so on.
