  Phylogenetic networks are generalizations of phylogenetic trees that allow
the representation of reticulation events such as horizontal gene transfer or
hybridization, and can also represent uncertainty in inference. A subclass of
these, tree-based phylogenetic networks, have been introduced to capture the
extent to which reticulate evolution nevertheless broadly follows tree-like
patterns. Several important operations that change a general phylogenetic
network have been developed in recent years, and are important for allowing
algorithms to move around spaces of networks; a vital ingredient in finding an
optimal network given some biological data. A key such operation is the Nearest
Neighbor Interchange, or NNI. While it is already known that the space of
unrooted phylogenetic networks is connected under NNI, it has been unclear
whether this also holds for the subspace of tree-based networks. In this paper
we show that the space of unrooted tree-based phylogenetic networks is indeed
connected under the NNI operation. We do so by explicitly showing how to get
from one such network to another one without losing tree-basedness along the
way. Moreover, we introduce some new concepts, for instance ``shoat networks'',
and derive some interesting aspects concerning tree-basedness. Last, we use our
results to derive an upper bound on the size of the space of tree-based
networks.
