Theorem 1
Let T' and T''
be any two crossing-free spanning trees of S.
Then T' can be transformed into
T'' by O(n^{2}) edge slides.
Theorem 2
Let T={p_{1}-p_{2}-...-p_{n}}
for odd n>2 have the shape of a spiral
numbered from inside to outside.
Then we need at least (n-1)(n-2)/2
edge slides to get a tree
containing the edge (p_{n-2},p_{n}).
By induction we need at least (n-3)(n-4)/2
edge slides to get
the edge (p_{n-4},p_{n-2}).
The yellow face contains n-3 inner edges
=> 2n-5 moves necessary.