Class of directed graph for which there is only one path to a given parent? From a nomenclature standpoint, I am wondering if there is a name for a class of directed graph that has only one path to any given parent.

I can visualize this shape as an upside down tree that may or may not have a root, kind of like this:



In looking at this, I can see it is not really an upside down tree.

Is there a name for this type of graph? Again the rules are:


graph is directed
there are no cycles
nodes can share children
no node can have multiple paths to the same parent


Note that these rules work the same in both directions. For example, if we take the graph shown above and reverse the direction of all the arrows, then the rules are still satisfied. What is the name of this thing?
An orientation such that for every $u-v$ path there exists a $v-u$ path is called a strong orientation. Conversely, an orientation that only has either a $u-v$ or a $v-u$ path is a weak orientation.
An orientation such that for every $u-v$ path there exists a $v-u$ path is called a strong orientation. Conversely, an orientation that only has either a $u-v$ or a $v-u$ path is a weak orientation.