  We describe a technique to reorganize topologies of Steiner trees by
exchanging neighbors of adjacent Steiner points. We explain how to use the
systematic way of building trees, and therefore topologies, to find the correct
topology after nodes have been exchanged. Topology reorganizations can be
inserted into the enumeration scheme commonly used by exact algorithms for the
Euclidean Steiner tree problem in $d$-space, providing a method of improvement
different than the usual approaches. As an example, we show how topology
reorganizations can be used to dynamically change the exploration of the usual
branch-and-bound tree when two Steiner points collide during the optimization
process. We also turn our attention to the erroneous use of a pre-optimization
lower bound in the original algorithm and give an example to confirm its usage
is incorrect. In order to provide numerical results on correct solutions, we
use planar equilateral points to quickly compute this lower bound, even in
dimensions higher than two. Finally, we describe planar twin trees, identical
trees yielded by different topologies, whose generalization to higher
dimensions could open a new way of building Steiner trees.
