  For a set $P$ of $n$ points in the plane in general position, a non-crossing
spanning tree is a spanning tree of the points where every edge is a
straight-line segment between a pair of points and no two edges intersect
except at a common endpoint. We study the problem of reconfiguring one
non-crossing spanning tree of $P$ to another using a sequence of flips where
each flip removes one edge and adds one new edge so that the result is again a
non-crossing spanning tree of $P$. There is a known upper bound of $2n-4$ flips
[Avis and Fukuda, 1996] and a lower bound of $1.5n - 5$ flips. We give a
reconfiguration algorithm that uses at most $2n-3$ flips but reduces that to
$1.5n-2$ flips when one tree is a path and either: the points are in convex
position; or the path is monotone in some direction. For points in convex
position, we prove an upper bound of $2d - \Omega(\log d)$ where $d$ is half
the size of the symmetric difference between the trees. We also examine whether
the happy edges (those common to the initial and final trees) need to flip, and
we find exact minimum flip distances for small point sets using exhaustive
search.
