graph theory
On Wikipedia, it states an arborescence is a digraph for which a vertex $u$ called the root and any other vertex $v$, there is exactly one directed path from $u$ to $v$.
In other words, does that mean an arborescence is a digraph whose underlying graph is a tree?
As suggested by the terminology, any strongly connected graph is weakly connected, but a weakly connected graph is not necessarily strongly connected. For instance, the graph $1 \to 2$ is weakly connected but is not strongly connected.
It is easy to see that having indegree, outdegree $\ge 1$ for every vertex is a necessary condition for a graph to be strongly connected. Otherwise, you would have an unreachable vertex or a vertex from which there are no paths.
But it is not a sufficient condition; counterexample below.
Draw two directed triangle graphs and join them by a single directed edge.
Since the degrees are not balanced, there is a directed path from one strongly connected component to the other, but not the other way around; despite the underlying undirected graph being connected.
Best Answer
In fact, more is true. Given any tree, every edge can be given uniquely a direction that is towards the root of the tree. In other words, any tree can be uniquely converted into an arborescence. Conversely, any arborescence can be uniquely converted into a tree by changing each directed edge into an undirected edge.
EDIT: We always assume that there is given a root with the tree or arborescence.