One definition of <a href="https://en.wikipedia.org/wiki/Tree_(graph_theory)">tree</a> in graph theory could be as follows:
>A tree is a(n undirected) graph for which there is a unique (undirected) <a href="https://en.wikipedia.org/wiki/Path_(graph_theory)">path</a> between any pair of vertices.

This suggest a possible definition of "directed tree":
> A "directed tree" is a directed graph for which there is a unique <a href="https://en.wikipedia.org/wiki/Path_(graph_theory)">directed path</a> between any pair of vertices.
 

**Question:** Is there an established name for the "directed trees" defined above?

Note that these "directed trees" are *not* <a href="https://en.wikipedia.org/wiki/Arborescence_(graph_theory)">arborescences</a> (rooted directed trees). For example, a directed cycle is a "directed tree" in the above sense; and indeed all "directed trees" in the above sense are basically trees of directed cycles.