Is this graph a tree?
I'm not sure if my answer and especially my reasoning is correct?
A tree is an undirected graph where two arbitrary vertices are connected by exactly one path, e.g. a graph is a tree if it is acyclic and connected. A directed graph is a tree if its undirected form is a tree. So if we consider the graph as an undirected tree, we have that it's cyclic. As example, we have the path $(A,C,D,A)$. Since the undirected graph is not acyclic, it is not a tree. Since the undirected graph is not a tree, its directed form is not a tree. Thus the given directed graph above is not a tree.
Is my reasoning correct and is there an easier / better way to reason it?