We study the decidability and computational complexity for several decision problems related to limited nondeterminism of finite-state automata equipped with a pushdown stack. Ambiguity and tree width are two measures of nondeterminism considered in the literature. As a main result, we prove that the problem of deciding whether or not the tree width of a nondeterministic pushdown automaton is finite is decidable in polynomial time. We also prove that the k-tree width problem for nondeterministic input-driven pushdown automata (respectively, nondeterministic finite automata) is complete for exponential time (respectively, for polynomial space).