I see your point about it not being a strict tree. It is tree-like, however, in the sense that the branches/edges only point in one direction: forward in time.
The key property of real trees is that they _branch_ and the branches don't recombine.
I do agree that branches staying separate is the essential property of a tree data structure.
The need to connect leaves to roots is what gives individuals trees their branching nature. Obviously a tree can have holes in it etc, but loops in nutrient delivery present an inherent issue.