  An improved translation from alternating parity automata on infinite words to
alternating weak automata is given. The blow-up of the number of states is
related to the size of the smallest universal ordered trees and hence it is
quasi-polynomial, and only polynomial if the asymptotic number of priorities is
logarithmic in the number of states. This is an exponential improvement on the
translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on
the translation of Boker and Lehtinen (2018). Any slightly better such
translation would (if---like all presently known such translations---it is
efficiently constructive) lead to algorithms for solving parity games that are
asymptotically 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 it
would yield a significant breakthrough.
