Generalization of plane geometric trees? View a plane tree drawn in $\mathbb{R}^2$ as a joining of geometric (straight) segments at endpoints such that (a) they avoid intersecting one another (except where they share a vertex), and (b) they avoid creating a cycle, which would enclose a positive planar area.
I am interested in the generalization to $\mathbb{R}^3$ as follows.
Join together (flat) polygons, glued edge-to-edge, such that (a) they avoid intersecting one another (except where they share vertices and/or edges), and (b) they avoid enclosing (water-tightly) 
a positive volume.
My question is:

Is there a name for this construct?  Has it been studied?

I am mainly seeking references to any literature on this or related concepts.
Of course there is a generalization to $\mathbb{R}^d$, but I would be happy to learn
of work just generalizing plane trees to ??? in $\mathbb{R}^3$.
I cannot think of what it might be named: open panel structures?
It's come up in my work, and I would be delighted to christen it, but surely it has been
studied...?
Thanks in advance!
Addendum. As Greg Kuperberg kindly explained, the concept I described is
a collapsible complex.  It is usually defined for simplicial complexes, but works
as well when the constituents are polytopes rather than simplices, e.g., polygons in $\mathbb{R}^3$.
 A: The standard name for such a subset of $\mathbb{R}^3$ is a contractible (compact) polyhedron if, in addition to your criteria, you also demand that it be simply connected.  Of course you can think about contractible polyhedra of any dimension, not just 1 or 2, and embedded in any Euclidean space, or not embedded at all.
But, after choosing that name, there is a big surprise that was discovered by Bing and Borsuk.  Namely, that a contractible 2-dimensional polyhedron doesn't have to have a free edge!  Bing's version of it is called the house with two rooms, and it is the object on the left:

(The other part of the diagram shows how you can fill the house with bricks to prove that a ball collapses onto it.)
To get something more tree-like, you need to impose more restrictive conditions.  Associated with the examples of Bing and Borsuk, a polyhedron is called collapsible if it is either a disk, or a simpler collapsible polyhedron with a disk attached along some embedded edge.  Collapsible polyhedra are one type of more strictly tree-like polyhedra, and I guess that they have been studied since they have a name.
A: Also check out the House with One Room notes form Allen Hatcher
