**O. Aichholzer, F. Aurenhammer, and F. Hurtado**

Let be the set of all crossing-free spanning trees of a planar
-point set . We prove that contains, for each of its
members , a length-decreasing sequence of trees
such that
, , does not cross for , and
. Here denotes the Euclidean minimum spanning tree of
the point set . As an implication, the number of length-improving and
planar edge moves needed to transform a tree
into
is only . Moreover, it is possible to transform any two trees in
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.