If you have a tree that also might merge branches, but only in a directed way, i.e. all edges are one step either towards or away from the "root", what can we call it? It's a special case of a directed graph.
Version control systems with branches such as git have these graph structures. In git you have the forward/backward in time (with the epoch as the "root") with the possibility of branches being merged.
Example from the git page on wikipedia, 1 being the "root":
( https://commons.wikimedia.org/wiki/File:Revision_controlled_project_visualization-2010-24-02.svg )