  Let $\Gamma$ be a rooted tree and let $t$ be a positive integer. We study
algebraic invariants and properties of the path ideal generated by monomial
corresponding to paths of length $(t-1)$ in $\Gamma$. In particular, we give a
recursive formula to compute the graded Betti numbers, a general bound for the
regularity, an explicit computation of the linear strand, and we characterize
when this path ideal has a linear resolution.
