Before revision:  In particular, by plugging in the algorithm of M\k{a [ FOCS 2013 ], we obtain an O(n^{ 10 / 7 }) time algorithm for computing the permutation distance between two trees on n nodes.
After revision:  In particular, by plugging in the algorithm of Liu and Sidford [ FOCS 2013 ], we obtain an O(n^{ 10 / 7 }) time algorithm for computing the permutation distance between two trees on n nodes.