AbstractHalin [2] has shown that primitive sets with respect to a subset A of the vertex set of a connected graph G form a complete lattice (Halin-lattice). In this article special contractions are defined such that pairs (G, A) and these maps form a category HG and that a contravariant functor exists from HG to the category of complete lattices and lattice homomorphisms. Using this functor it is proved that the lattice homomorphism is a well-quasi ordering in the class of all Halin-lattices of rooted trees