In this paper, we prove that if G is a group acting on a tree X without inversions such that for every vertex v of X, and every edge e of X, the vertex group  of v is finitely generated having an infinite cyclic subgroup of finite index, the edge group  of e being finite, and the quotient graph  for the action of G on X being finite, then Gis finitely generated having an infinite cyclic subgroup of finite index.

We have applications to tree product of the groups and HNN extension groups.

groups acting on trees, ends of groups, tree product of groups, HNN groups.