I am realizing why I was having trouble finding information on this. I have been approaching the topic from a graph theory point of view while typically this type of concept is found when considering dynamical systems. Looking at limit sets, stable and unstable sets, etc., it appears that my "rooted tree at x" is similar to a stable set, only I am not requiring my space to be a topological space, I am not requiring f to be a homomorphism, and I am not terribly concerned with fixed points. I guess my naming convention is as good as any. If anyone has a suggestion on a decent book that might help me grasp the concept as it arises in dynamical systems, I would be grateful. Of course, I will be searching for materials on my own if no one has any ideas for me.