  We study the positive theory of groups acting on trees and show that under
the presence of weak small cancellation elements, the positive theory of the
group is trivial, i.e. coincides with the positive theory of a non-abelian free
group. Our results apply to a wide class of groups, including non-virtually
solvable fundamental groups of $3$-manifold groups, generalised
Baumslag-Solitar groups and almost all one-relator groups and graph products of
groups. It follows that groups in the class satisfy a number of algebraic
properties: for instance, their verbal subgroups have infinite width and,
although some groups in the class are simple, they cannot be boundedly simple.
  In order to prove these results we describe a uniform way for constructing
(weak) small cancellation tuples from (weakly) stable elements. This result of
interest in its own is fundamental to obtain corollaries of general nature such
as a quantifier reduction for positive sentences or the preservation of the
non-trivial positive theory under extensions of groups.
