<p>Graham, Lovász and Pollak obtained a well known formula for the determinant of distance matrices of trees. This formula depends only on the number of vertices of the tree and not on its topological structure. Later, Hou and Woo computed the explicit expression of the Smith form of the distance matrix of a tree, which again, depends only on the number of vertices. Here, we extend Graham-Lovász-Pollak and Hou-Woo results to higher dimensions by computing the Smith normal form and the determinant of a distance matrix associated to high dimensional trees.</p>