  A group is boundedly simple if, for some constant N, every nontrivial
conjugacy class generates the whole group in N steps. For a large class of
trees, Tits proved simplicity of a canonical subgroup of the automorphism
group, which is generated by pointwise stabilizers of edges. We prove that only
for uniform subdivisions of biregular trees are such groups boundedly simple.
In fact these groups are 8-boundedly simple. As a consequence, we prove that if
G is boundedly simple (or from a certain class K) and G acts by automorphisms
on a tree, then G fixes some vertex of A, or stabilizes some end of A, or the
smallest nonempty G-invariant subtree of A is a uniform subdivision of a
biregular tree.
