What is the name of a rooted tree whose nodes may have edges to their descendants? A tree is a special kind a graph.

However, I came across a data structure which is a like a rooted tree, but where nodes are authorized to have direct links to any of their descendants. Shortcuts if you will.

This is not a tree anymore.

This is a specialized DAG that has more restriction. That is, it has a single root (or source)

Does this type of graph have a name?

CLARIFICATION:

By 'tree', I'm referring to the tree data structure, not a tree in graph theory. It seems crazy to me that such related objects have the same name even though they don't mean the same thing.
The tree data structure is always rooted but there are rooted DAG that are not trees.

{1→2, 2→3, 1→3, 4→2} doesn't qualify because 4→2 is an edge toward an ancestor, not a descendant.
{1→2, 1→3, 2→4, 3→4} doesn't qualify.

{1→2, 2→3, 1→3} and {1→2, 1→3, 2→4, 3→5, 3→6, 1→4, 1→6} both qualify.
I'm not aware of any standard term. The graphs that you describe are subgraphs of the transitive closure of a tree, if that's of any help to you, but one wouldn't want to use that phrase twenty times in a paper.

But... as Pål GD alludes to in a comment, every DAG on $n$&nbsp;vertices is a subgraph of the transitive closure of the $n$-vertex directed path. So this answer is a bit of a fail, really.