Growth in higher Baumslag-Solitar groups

We study the HNN extension of given by the cubing endomorphism , and prove that such groups have rational growth with respect to the standard generating sets. We compute the subgroup growth series of the horocyclic subgroup in this family of examples, prove that for each m the subgroup has rational growth. We then use the tree-like structure of these groups to see how to compute the growth of the whole group.