  Let $R=k[x_{1},\ldots,x_{n}]$, where $k$ is a field. The path ideal (of
length $t\geq 2$) of a directed graph $G$ is the monomial ideal, denoted by
$I_{t}(G)$, whose generators correspond to the directed paths of length $t$ in
$G$. Let $\Gamma$ be a directed rooted tree. We characterize all such trees
whose path ideals are unmixed and Cohen-Macaulay.
  Moreover, we show that $R/I_{t}(\Gamma)$ is Gorenstein if and only if the
Stanley-Reisner simplicial complex of $I_{t}(\Gamma)$ is a matroid.
