  We prove that, in all dimensions d>=4, every simple open polygonal chain and
every tree may be straightened, and every simple closed polygonal chain may be
convexified. These reconfigurations can be achieved by algorithms that use
polynomial time in the number of vertices, and result in a polynomial number of
``moves.'' These results contrast to those known for d=2, where trees can
``lock,'' and for d=3, where open and closed chains can lock.
