  The path ideal (of length t >=2) of a graph G is the monomial ideal, denoted
I_t(G), whose generators correspond to the directed paths of length t in G. We
study some of the algebraic properties of I_t(G) when G is a tree. We first
show that I_t(G) is the facet ideal of a simplicial tree. As a consequence, the
quotient ring R/I_t(G) is always sequentially Cohen-Macaulay, and the Betti
numbers of R/I_t(G) do not depend upon the characteristic of the field. We
study the case of the line graph in greater detail at the end of the paper.
