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Schmidt characterised the class of rayless graphs by an ordinal rankfunction, which makes it possible to prove statements about rayless graphs bytransfinite induction. Halin asked whether Schmidt's rank function can begeneralised to characterise other important classes of graphs. We answerHalin's question in the affirmative. Another largely open problem raised byHalin asks for a characterisation of the class of graphs with an end-faithfulspanning tree. A well-studied subclass is formed by the graphs with a normalspanning tree. We determine a larger subclass, the class of normally traceablegraphs, which consists of the connected graphs with a raylesstree-decomposition into normally spanned parts. Investigating the class ofnormally traceable graphs further we prove that, for every normally traceablegraph, having a rayless spanning tree is equivalent to all its ends beingdominated. Our proofs rely on a characterisation of the class of normallytraceable graphs by an ordinal rank function that we provide.

End-faithful spanning trees in graphs without normal spanning trees