The Calkin-Wilf tree of a quadratic surd

By using the Calkin-Wilf tree, we prove the irrationality of numbers of the form $\alpha =\frac{\sqrt{N}+p}{q}$ where $N$ is a positive integer which is not a perfect square, $p$ is a rational integer such that $p^2<N$ and $q$ is a positive integer which divides $N-p^2$. For this, we consider an analogue of the Calkin-Wilf tree with root $\alpha$ and we define a special path in this tree which satisfies remarkable properties of periodicity and symmetry. This path is closely related to the continued fraction expansion of $\alpha$ and allows us to give new proofs of theorems due to Legendre and to Galois about the form of such an expansion in special cases of square roots and reduced quadratic surds.