  Let $G$ be a graph and $T_1,T_2$ be two spanning trees of $G$. We say that
$T_1$ can be transformed into $T_2$ via an edge flip if there exist two edges
$e \in T_1$ and $f$ in $T_2$ such that $T_2= (T_1 \setminus e) \cup f$. Since
spanning trees form a matroid, one can indeed transform a spanning tree into
any other via a sequence of edge flips, as observed by Ito et al.
  We investigate the problem of determining, given two spanning trees $T_1,T_2$
with an additional property $\Pi$, if there exists an edge flip transformation
from $T_1$ to $T_2$ keeping property $\Pi$ all along.
  First we show that determining if there exists a transformation from $T_1$ to
$T_2$ such that all the trees of the sequence have at most $k$ (for any fixed
$k \ge 3$) leaves is PSPACE-complete.
  We then prove that determining if there exists a transformation from $T_1$ to
$T_2$ such that all the trees of the sequence have at least $k$ leaves (where
$k$ is part of the input) is PSPACE-complete even restricted to split,
bipartite or planar graphs. We complete this result by showing that the problem
becomes polynomial for cographs, interval graphs and when $k=n-2$.
