Diestel and Leader have characterised connected graphs that admit a normal spanning tree via two classes of forbidden minors. One class is Halin's (aleph(0), aleph(1))-graphs: bipartite graphs with bipartition (A, B) such that vertical bar A vertical bar = aleph(0), vertical bar B vertical bar = aleph(1) and every vertex of B has infinite degree. Our main result is that under Martin's Axiom and the failure of the Continuum Hypothesis, the class of forbidden (aleph(0), aleph(1))-graphs in Diestel and Leader's result can be replaced by one single instance of such a graph. Under CH, however, the class of (aleph(0), aleph(1))-graphs contains minor-incomparable elements, namely graphs of binary type, and U-indivisible graphs. Assuming CH, Diestel and Leader asked whether every (aleph(0), aleph(1))-graph has an (aleph(0), aleph(1))-minor that is either indivisible or of binary type, and whether any two U-indivisible graphs are necessarily minors of each other. For both questions, we construct examples showing that the answer is in the negative.