We consider the problems of unfolding 3D lattice polygons embedded on the surface of some classes of lattice polyhedra, and of unfolding 2D orthogonal trees. During the unfolding process, all graph edges are preserved and no edge crossings are allowed. Let  n be the number of edges of the given polygon or tree. We show that a lattice polygon embedded on an open lattice orthotube can be convexified in  O ( n ) moves and time, and a lattice polygon embedded on a lattice Tower of Hanoi, a lattice Manhattan Tower, or an orthogonally-convex lattice polyhedron can be convexified in  O ( n 2) moves and time. The main technique in our algorithms is to fold up the lattice polygon from the end blocks of the given lattice polyhedron. On the other hand, we show that a 2-monotone orthogonal tree on the plane can be straightened in  O ( n 2) moves and time. We hope that our results shed some light on solving the more general conjectures, which we proposed, that a 3D lattice polygon embedded on any lattice polyhedron can always be convexified, and any 2D orthogonal tree can always be straightened.
['Sheung-Hung Poon']
On Unfolding 3D Lattice Polygons and 2D Orthogonal Trees