  We investigate the collapsibility of systolic finite simplicial complexes of
arbitrary dimension. The main tool we use in the proof is discrete Morse
theory. We shall consider a convex subcomplex of the complex and project any
simplex of the complex onto a ball around this convex subcomplex. These
projections will induce a convenient gradient matching on the complex. Besides
we analyze the combinatorial structure of both CAT(0) and systolic locally
finite simplicial complexes of arbitrary dimensions. We will show that both
such complexes possess an arborescent structure. Along the way we make use of
certain well known results regarding systolic geometry.
