  The geometric $\delta$-minimum spanning tree problem ($\delta$-MST) is the
problem of finding a minimum spanning tree for a set of points in a normed
vector space, such that no vertex in the tree has a degree which exceeds
$\delta$, and the sum of the lengths of the edges in the tree is minimum. The
similarly defined geometric $\delta$-minimum bottleneck spanning tree problem
($\delta$-MBST), is the problem of finding a degree bounded spanning tree such
that the length of the longest edge is minimum. For point sets that lie in the
Euclidean plane, both of these problems have been shown to be NP-hard for
certain specific values of $\delta$. In this paper, we investigate the
$\delta$-MBST problem in $3$-dimensional Euclidean space and $3$-dimensional
rectilinear space. We show that the problems are NP-hard for certain values of
$\delta$, and we provide inapproximability results for these cases. We also
describe new approximation algorithms for solving these $3$-dimensional
variants, and then analyse their worst-case performance.
