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An improved translation from alternating parity automata on infinite words toalternating weak automata is given. The blow-up of the number of states isrelated to the size of the smallest universal ordered trees and hence it isquasi-polynomial, and only polynomial if the asymptotic number of priorities islogarithmic in the number of states. This is an exponential improvement on thetranslation of Kupferman and Vardi (2001) and a quasi-polynomial improvement onthe translation of Boker and Lehtinen (2018). Any slightly better suchtranslation would (if---like all presently known such translations---it isefficiently constructive) lead to algorithms for solving parity games that areasymptotically faster in the worst case than the current state of the art(Calude, Jain, Khoussainov, Li, and Stephan, 2017; Jurdzi\'nski and Lazi\'c,2017; and Fearnley, Jain, Schewe, Stephan, and Wojtczak, 2017), and hence itwould yield a significant breakthrough.

Alternating Weak Automata from Universal Trees