  The theory of complexes of directed trees was initiated by Kozlov to answer a
question by Stanley, and later on, results from the theory were used by Babson
and Kozlov in their proof of the Lovasz conjecture. We develop the theory and
prove that complexes on directed acyclic graphs are shellable.
  A related concept is that of independence complexes: construct a simplicial
complex on the vertex set of a graph, by including each independent set of
vertices as a simplex. Two theorems used for breaking and gluing such complexes
are proved and applied to generalize results by Kozlov.
  A fruitful restriction is anti-Rips complexes: a subset P of a metric space
is the vertex set of the complex, and we include as a simplex each subset of P
with no pair of points within distance r. For any finite subset P of \mathbb{R}
the homotopy type of the anti-Rips complex is determined.
