I will give a new definition of partial Higher Dimension Automata – a geometric model for true concurrency – using lax functors. This definition is simpler than the original, and more natural from a categorical point of view, but also matches more clearly the intuition that pHDA are Higher Dimensional Automata with some missing faces. I will then focus on trees. Originally, for example in transition systems, trees are defined as those systems that have a unique path property. To understand what kind of unique path property is needed in pHDA, I will start by looking at trees as colimits of paths. From this, I will deduce that trees are exactly the pHDA with the unique path property modulo a notion of homotopy, and without any shortcuts. This property will allow me to prove two interesting characterisations of trees: trees are exactly those pHDA that are the unfolding of another pHDA; trees are exactly the cofibrant objects, much as in the language of Quillen’s model structures. In particular, this last characterisation gives the premisses of a new understanding of concurrency theory using homotopy theory.