  In this series we introduce and investigate the concept of connectoids, which
captures the connectivity structure of various discrete objects like undirected
graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids.
  In the first paper of this series we developed a universal end space theory
based on connectoids that unifies the existing end spaces of undirected and
directed graphs. Further, we introduced a notion of normal trees of connectoids
which represents the ends of connectoids as normal trees do in undirected
graphs.
  In this paper we show that our notion of normal trees generalises normal
trees of undirected graphs to connectoids by characterising their existence: We
extend Jung's famous characterisation via dispersed sets to connectoids, and
show that normal spanning trees exist if they exist in some neighbourhood of
each end. Furthermore, we show that a connectoid has a normal spanning tree if
and only if its groundset can be well-ordered in a certain way, called
countable separation number.
  In this way we establish a unified notion of normal trees for undirected,
directed and bidirected graphs, as well as for hypergraphs and finitary
matroids.
