A quadratic distance bound on sliding between crossing-free spanning trees

Let S be a set of n points in the plane and let TS be the set of all crossing-free spanning trees of S. We show that any two trees in TS can be transformed into each other by O(n2) local and constant-size edge slide operations. No polynomial upper bound for this task has been known, but in O.Aichholzer, F.Aurenhammer, F.Hurtado Sequences of spanning trees and a fixed tree theorem. Computational Geometry: Theory and Applications, 21(1-2):3-20, 2002. a bound of O(n2 log n) operations was conjectured.