In this paper we show that an affine bijection f: T-1 --> T-2 between two p
olyhedral complexes T-1, T-2, both of which consist of a union of faces of
two convex polyhedra P-1 and P-2, necessarily respects the cell-complex str
ucture of T-1 and T-2 inherited from P-1 and P-2, respectively, provided f
extends to an affine map from P-1 into P-2. In addition, we present an appl
ication of this result within the area of T-theory to obtain a far-reaching
generalization of previous results regarding the equivalence of two distin
ct constructions of the phylogenetic tree associated to "perfect" (that is,
treelike) distance data.