\begin{definition}[Definition:Rooted Subtree]
Let $\left({T, r_T}\right)$ be a rooted tree.
A '''rooted subtree''' of $T$ is a rooted tree $\left({S, r_S}\right)$ such that:
:$S$ is a subtree of $T$
:$r_S = r_T$
Note that the second condition implies that $r_T \in S$.
\end{definition}