We study the descriptional cost of removing nondeterminism in constant height pushdown automata -i.e., pushdown automata with a built-in constant limit on the height of the pushdown. We show a double-exponential size increase when converting a constant height nondeterministic pushdown automaton into an equivalent deterministic device. Moreover, we prove that such a double-exponential blow-up cannot be avoided by certifying its optimality. As a direct consequence, we get that eliminating

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... minism in classical finite state automata is single-exponential even with the help of a constant height pushdown store.