  We study a notion of deformation for simplicial trees with group actions
(G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a
deformation if there is a finite sequence of collapse and expansion moves
joining them. We show that this relation on the set of G-trees has several
characterizations, in terms of dynamics, coarse geometry, and length functions.
Next we study the deformation space of a fixed G-tree X. We show that if X is
`strongly slide-free' then it is the unique reduced tree in its deformation
space.
  These methods allow us to extend the rigidity theorem of Bass and Lubotzky to
trees that are not locally finite. This yields a unique factorization theorem
for certain graphs of groups. We apply the theory to generalized
Baumslag-Solitar groups and show that many have canonical decompositions. We
also prove a quasi-isometric rigidity theorem for strongly slide-free G-trees.
