Non-forking and preservation of NIP and dp-rank

We investigate the question of whether restriction an NIP type p?S(B) which does not fork over A?B to A is also NIP, and analogous for dp-rank. show that if B contains a Morley sequence I generated by p A, then p?AI similarly preserves This yields positive answers generically stable types case types. With similar techniques we provide new more direct proof latter. Moreover, introduce general construction “trees whose open cones are models some theory” in particular inp-minimal theory DTR dense trees with random graphs on cones, exemplifies negative answer question.