Abstract: The atomic cluster expansion (ACE) provides basis functions that are complete in the space of atomic environments. The basis functions depend on the configuration, the difference vectors between atoms that are used to characterize the structure up to a global translation. The most important configuration of ACE corresponds to a star graph, while in general admissible configurations are trees. The star graph appears natural for modeling local interactions and further trees incorporate semilocal interactions. By employing basis functions that are products of single bond basis functions and by tensor decomposing corresponding expansion coefficients, it becomes possible to sum over possible tree configurations. The resulting iterative procedure is efficient to evaluate and derives current message passing potentials. In my presentation if will start by introducing local ACE and compare and contrast it to global, graph ACE. I will then show how trees including their subtrees can be taken into account and evaluated efficiently. This will also enable basic estimates of the numerical cost of incorporating semilocal interactions. I will end with numerical illustrations.