Alternating automata play a key role in the automata-theoretic approach to specification, verification, and synthesis of reactive systems. Many algorithms on alternating automata, and in particular, their nonemptiness test, involve removal of alternation: a translation of the alternating automaton to an equivalent nondeterministic one. For alternating Büchi automata, the best known translation uses the "breakpoint construction" and involves an O(3 n ) state blow-up. The translation was described by Miyano and Hayashi in 1984, and is widely used since, in both theory and practice. Yet, the best known lower bound is only 2 n . In this paper we develop and present a complete picture of the problem of alternation removal in alternating Büchi automata. In the lower bound front, we show that the breakpoint construction captures the accurate essence of alternation removal, and provide a matching Ω(3 n ) lower bound. Our lower bound holds already for universal (rather than alternating) automata with an alphabet of a constant size. In the upper-bound front, we point to a class of alternating Büchi automata for which the breakpoint construction can be replaced by a simpler n2 n construction. Our class, of ordered alternating Büchi automata, strictly contains the class of very-weak alternating automata, for which an n2 n construction is known.