An example of an end-type action on a Λ-tree

Let Λ be an ordered abelian group. We give an example of a non-trivial action of end type of a group G on a Λ-tree which is faithful and transitive and such that G is not soluble and contains no non-abelian free subgroup. The degree of each point is 3. This is related to certain automorphisms of the infinite rooted binary tree.