how to define a "directed spanning tree"?

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In all my books and articles about "graph theory", I didn't find the definition of "directed spanning tree". Could you please give this definition and the reference?

How to judge if a directed graph is a directed spanning tree?

Thanks.

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As was mentioned arborescence is likely what you are looking for.

A graph is a spanning tree if it is a tree (acyclic, connected graph) that touches each node.

In Directed Spanning trees it looks like either you choose a node, mark it as the root and build a tree that is defined as being a single path from that node to each other node.