  Parity games have witnessed several new quasi-polynomial algorithms since the
breakthrough result of Calude et al. (STOC 2017). The combinatorial object
underlying these approaches is a universal tree, as identified by Czerwi\'nski
et al. (SODA 2019). By providing a quasi-polynomial lower bound on the size of
universal trees, they have highlighted a barrier that must be overcome by all
existing approaches to attain polynomial running time. This is due to the
existence of worst case instances which force these algorithms to explore a
large portion of the tree.
  As an attempt to overcome this barrier, we propose a strategy iteration
framework which can be applied on any universal tree. It is at least as fast as
its value iteration counterparts, while allowing one to take bigger leaps in
the universal tree. Our main technical contribution is an efficient method for
computing the least fixed point of 1-player games. This is achieved via a
careful adaptation of shortest path algorithms to the setting of ordered trees.
By plugging in the universal tree of Jurdzi\'nski and Lazi\'c (LICS 2017), or
the Strahler universal tree of Daviaud et al. (ICALP 2020), we obtain
instantiations of the general framework that take time $O(mn^2\log n\log d)$
and $O(mn^2\log^3 n \log d)$ respectively per iteration.
