Math. Systems Theory 29, 411-421 (1996) Mathematical Systems Theory 9 1996 Springer-Verlag New York Inc. On Balanced Versus Unbalanced Computation Trees* U. Hertrampf, H. Vollmer, and K. W. Wagner Theoretische Informatik, Universit~it Wtirzburg, Am Exerzierplatz 3, D-97072 Wtirzburg, Germany {hertramp,vollmer, wagner}@ informatik.uni-wuerzburg.d e Abstract. A great number of complexity classes between P and PSPACE can be defined via leaf languages for computation trees of nondeterministic polynomial- time machines. Jenner, McKenzie, and Th6rien (Proceedings of the 9th Conference on Structure in Complexity Theory, 1994) raised the issue of whether considering balanced or unbalanced trees makes any difference. For a number of leaf-language classes, coincidence of both models was shown, but for the very prominent ex- ample of leaf-language classes from the alternating logarithmic-time hierarchy the question was left open. It was only proved that in the balanced case these classes exactly characterize the classes from the polynomial-time hierarchy. Here, we show that balanced trees apparently make a difference: In the unbalanced case, a class from the logarithmic-time hierarchy characterizes the corresponding class from the polynomial-time hierarchy with a PP-oracle. Along the way, we get an interesting normal form for PP computations. 1. Introduction Almost all natural complexity classes in the range between P and PSPACE can be defined by putting certain requirements on the computation trees produced by nondeterministic machines. Take for example the class NP, where we require that, for a certain machine M defining an NP-language, a word is in the language under consideration if and only if the computation tree possesses at least one accepting path. For (~) P, the number of accepting paths is required to be even. For PP, the number of accepting paths has to be greater than the number of rejecting paths. * The first and third authors were supported by Deutsche Forschungsgemeinschaft, Grant No. Wa 847/1-1, "k-wertige Schaltkreise." The second author was supported in part by an Alexander yon Humboldt fellowship.