  We investigate path-, ray- and branch spaces of trees, certain topological
spaces naturally associated with order theoretic trees, and provide topological
characterisations for these spaces in terms of the existence of certain kinds
of (sub-)bases. Our results yield a solution to a problem due to Diestel from
1992, by establishing the following equivalences for any Hausdorff space $X$:
  (1) $X$ is homeomorphic to the end space of a graph,
  (2) $X$ is homeomorphic to the ray space of a special order tree,
  (3) $X$ admits a nested clopen subbase that is noetherian, hereditarily
complete and $\sigma$-disjoint.
