We analyze the relationship between three ways of generating trees using collapsible pushdown systems (CPSu27s): using deterministic CPSu27s, nondeterministic CPSu27s, and deterministic word-accepting CPSu27s. We prove that (for each level of the CPS and each input alphabet) the three classes of trees are equal. The nontrivial translations increase n-1 times exponentially the size of the level-n CPS. The same results stay true if we restrict ourselves to higher-order pushdown systems without collapse. As a second contribution we prove that the hierarchy of word languages recognized by nondeterministic CPSu27s is infinite. This is a consequence of a lemma which bounds the length of the shortest accepting run. It also implies that the hierarchy of epsilon-closures of configuration graphs is infinite (which was already known). As a side effect we obtain a new algorithm for the reachability problem for CPSu27s; it has the same complexity as previously known algorithms.
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