A constructive approach to minimal free resolutions of path ideals of trees

For a rooted tree Ґ; we consider path ideals of Ґ, which are ideals that are generated by all directed paths of a fixed length in Ґ. In this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. In particular, we provide a class of subforests of Ґ that are in one-to-one correspondence with the multi-graded Betti numbers of the path ideal as well as providing a method for determining the projective dimension and the Castelnuovo-Mumford regularity of a given path ideal.