  The Dry Ten Martini Problem for Sturmian Hamiltonians is affirmatively
solved. Concretely, we prove that all spectral gaps are open for Schr\"odinger
operators with Sturmian potentials and non-vanishing coupling constant. A key
approach towards the solution is a representation of the spectrum as the
boundary of an infinite tree. This tree is constructed via particular periodic
approximations and it encodes substantial spectral characteristics.
