# Can a (directed) tree contain crossing lines?

Is the graph below a (directed) tree in the mathematical sense, and if it is, is there any precise definition that distinguishes crossing from non-crossig trees? I think it should be a tree since it is acyclic and connected, but the crossing edges $B - F$ and $C - E$ make me unsure about it - it doesn't look like a "classical" tree, intuitively (presumably because by "classical" tree, I have directed and rooted trees in mind, although I am aware that undirected trees exist, too), and I can not recall having seen such a graph referred to as 'tree' ever before.
Which graph-theoretical classification would apply to this object? Are the distinctions directed vs. undirected, and ordered vs. unordered of relevance here?

Edit: I am thinking that ordering might be of relevance because if you were to "untangle" the lines and align the nodes in such a way that node $F$ is right next to $D$ and $E$ is right next to $G$, the linear order of nodes would be a different one, and that this would make it different tree. Is this correct?

• @lemontree No, the crossing doesn't happen on the level of the geometric realization. It happens when you map the realization to the plane. Graph (Vertices, Edges) -> Geometric Realization (Topological Space X) -> Drawing in plane (continuous map from f : X to R^2). For a well drawn graph, $f$ must be a topological embedding (meaning, injective and homeomorphic onto its image). You can detect crossings by studying $f$, for example, in your picture there is a point $p \in R^2$ with $f^{-1}(p)$ consisting of two points in the interior of distinct edges... – Lorenzo Najt Jan 26 '17 at 17:05