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 $σ$-disjoint.