How do we define a tree in a directed graph?

I am trying to solve the exercise 3.3 in Approximation Algorithms by Vazirani, pg 34. It states


3.3 Give an approximation factor preserving reduction from the set cover
problem to the following problem, thereby showing that it is unlikely to have a better approximation guarantee than $O(\log{n})$.


Problem 3.14 (Directed Steiner tree) $G = (V,E)$ is a directed graph
with nonnegative edge costs. The vertex set $V$ is partitioned into two sets, required and Steiner. One of the required vertices, $r$, is special. The problem is to find a minimum cost tree in $G$ rooted into $r$ that contains all the required vertices and any subset of the Steiner vertices.

My (related) questions

• How do we define a tree in a directed graph? I searched on the Internet and found that a tree is defined for only undirected tree, in particular on Wikipedia.

• What does it mean "rooted into $r$"? Does it mean out-degree of $r$ is zero?

Sometime there isn't a completely agreed upon meaning of terms, it is more useful to look at the context to see which definition is appropriate.

In this case, instead of searching for definition of general directed tree, it is better to look for directed steiner tree.

A quick search pulled up this paper which has this diagram showing an example of directed steiner tree:

While not explicitly defined, it coincides with my intuition of what a default directed tree should look like --- A directed graph with exactly one directed path from a distinguished root vertex to any other vertex in the graph (tree).

StackExchange Computer Science Q#85771, answer score: 8