Sentence:  We will say that the simplicial complex "X" is <WSD>"arborescent</WSD>" if we can obtain it by a sequence of Whitehead dilations starting from a point. This sequence can be "non"-locally finite. Also "<WSD>arborescent</WSD>" does "not" imply "collapsible" ("R2" for instance is <WSD>arborescent</WSD>). We will say that a 2-dimensional simplicial context "X" is <WSD>"almost arborescent</WSD>" if it can be obtained by a series of Whitehead dilations "and" of additions of 2-cells along their boundaries, starting from a point. 
Meaning of the word marked with <WSD> tags:  Like a tree; having a structure or appearance similar to that of a tree; branching.