We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta A, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of A and (b) does there exist a crisp-deterministic wta which computes the run semantics of A? We show that the finiteness of the Nerode algebra N (A) of A implies a positive answer for (a), and that the finite order property of A implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of N (A) and a sufficient condition which guarantees the finite order property of A. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if N (A) is finite, and similarly for (b) if A has finite order property. We prove that it is undecidable whether an arbitrary wta A is crisp-determinizable. We also prove that both, the finiteness of N (A) and the finite order property of A are undecidable.