  End-structures of infinite graphs sit at the interface between graph theory,
group theory and topology. They arise as the boundary of an infinite graph in a
standard sense generalising the theory of the Freudenthal boundary developed by
Freudenthal and Hopf in the 1940's for infinite~groups.
  A long-standing quest in infinite graph theory with a rich body of literature
seeks to represent or capture the end-structures of infinite graphs. Halin,
Diestel and Carmesin raised open questions that formalise this quest from
different perspectives. In this paper, we introduce a new decomposition
technique which allows us to systematically solve three of their open questions
in amended form, leading to a topological representation theorem showing that
every end space is homeomorphic to the end space of some (uniform graph on a)
special order tree.
