On the number of heterochromatic trees in nice and beautiful colourings of complete graphs

Motivated by the long-standing Graceful Tree Conjecture, we introduce classes of edge-colourings of the complete graph -- that we call nice and beautiful -- and study which and how many heterochromatic spanning trees appear under such colourings. We prove that if the colouring is nice, there is at least a quadratic number of different heterochromatic trees; and if the colouring is beautiful there is an exponential number of different such trees.