A class of spaces called the
S
U
V
∞
SU{V^\infty }
spaces has arisen in the study of a possibly noncompact variant of cellularity. These spaces play a role in this new theory analogous to that of the
U
V
∞
U{V^\infty }
spaces in cellularity theory. Herein it is shown that the locally compact metric space X is an
S
U
V
∞
SU{V^\infty }
space if and only if there exists a tree T such that X and T have the same proper shape. This result is then used to classify the proper shapes of the
S
U
V
∞
SU{V^\infty }
spaces, two such being shown to have the same proper shape if and only if their end-sets are homeomorphic. Also, a possibly noncompact analog of property
U
V
n
U{V^n}
, called
S
U
V
n
SU{V^n}
, is defined and it is shown that if X is a closed connected subset of a piecewise linear n-manifold, then X is an
S
U
V
n
SU{V^n}
space if and only if X is an
S
U
V
∞
SU{V^\infty }
space. Finally, it is shown that a locally finite connected simplicial complex is an
S
U
V
∞
SU{V^\infty }
space if and only if all of its homotopy and proper homotopy groups vanish.