The uniqueness of eigen-distribution under non-directional algorithms

Liu and Tanaka (2007) investigated the eigendistribution, which achieves the distributional complexity, for uniform binary trees. In the present work, we extend their studies to balanced multi-branching trees. We show that an eigen-distibution is equivalent to Eⁱ-distribution with respect to the closed set of all alpha-beta pruning algorithms. The proof is quite different from the uniform binary case given by Suzuki and Nakamura (2012). We also show that for any multibranching tree, Eⁱ-distribution is the unique eigen-distribution with respect to the set of all alpha-beta pruning algorithms.