  Let $1\leq r\leq n$ and suppose that, when the Depth-first Search Algorithm
is applied to a given rooted labelled tree on $n+1$ vertices, exactly $r$
vertices are visited before backtracking. Let $R$ be the set of trees with this
property. We count the number of elements of $R$.
  For this purpose, we first consider a bijection, due to Parkinson, Yang and
Yu, that maps $R$ onto the set of parking function with center (defined by the
authors in a previous article) of size $r$. A second bijection maps this set
onto the set of parking functions with run $r$, a property that we introduce
here. We then prove that the number of length $n$ parking functions with a
given run is the number of length $n$ rook words (defined by Leven, Rhoades and
Wilson) with the same run. This is done by counting related lattice paths in a
ladder-shaped region. We finally count the number of length $n$ rook words with
run $r$, which is the answer to our initial question.
