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We take a deeper dive into the geometry and the number theory that underlaythe butterfly graphs of the Harper and the generalized Harper models of Blochelectrons in a magnetic field. Root of the number theoretical characteristicsof the fractal spectrum is traced to a close relationship between the Fareytree -- the hierarchical tree that generates all rationals and the Wannierdiagram -- a graph that labels all the gaps of the butterfly graph. Theresulting Farey-Wannier hierarchical lattice of trapezoids provides geometricalrepresentation of the nested pattern of butterflies in the butterfly graph.Some features of the energy spectrum such as absence of some of the Wanniertrajectories in the butterfly graph fall outside the number theoreticalframework, can be stated as a simple rule of "minimal violation of mirrorsymmetry". In a generalized Harper model, Farey-Wannier representation prevailsas the lattice regroups to form some hexagonal unit cells creating new {\itspecies} of butterflies

Geometry, Number Theory and the Butterfly Spectrum of Two-Dimensional Bloch Electrons