A d-dimensional permutation is a sequence of d+1 permutations with the leading element being the identity permutation. It can be viewed as an alignment structure across d+1 sequences, or visualized as the result of permuting $n$ hypercubes of (d+1) dimensions. We study the hierarchical decomposition of d-dimensional permutations. We show that when d >= 2, the ratio between non-decomposable or simple d-permutations and all d-permutations approaches 1. We also prove that the growth rate of the number of d-permutations that can be factorized into k-ary branching trees approaches (k/e)^d as k grows.