Sequences of spanning trees and a fixed tree theorem

Let Ts be the set of all crossing-free spanning trees of a planar n-point set S. We prove that Ts contains, for each of its members T, a length-decreasing sequence of trees T0 ,..., Tk such that T0 = T, Tk = MST(S), Ti does not cross Ti-1 for i = 1,...,k, and k = O(logn). Here MST(S) denotes the Euclidean minimum spanning tree of the point set S.As an implication, the number of length-improving and planar edge moves needed to transform a tree T ∈ Ts into MST(S) is only O(n log n). Moreover, it is possible to transform any two trees in Ts into each other by means of a local and constant-size edge slide operation. Applications of these results to morphing of simple polygons are possible by using a crossing-free spanning tree as a skeleton description of a polygon.