  The hierarchical structure of the butterfly fractal -- the Hofstader
butterfly, is found to be described by an octonary tree. In this framework of
building the butterfly graph, every iteration generates sextuplets of
butterflies, each with a tail that is made up of an infinity of butterflies.
Identifying {\it butterfly with a tale} as the building block, the tree is
constructed with eight generators represented by unimodular matrices with
integer coefficients. This Diophantine description provides one to one mapping
with the butterfly fractal, encoding the magnetic flux interval and the
topological quantum numbers of every butterfly. The butterfly tree is a
generalization of the ternary tree describing the set of primitive Pythagorean
triplets.
