Areas-of-areas on Hall trees generate the shuffle algebra

We consider the coordinate-iterated-integral as an algebraic product on the shuffle algebra, called the (right) half-shuffle product. Its anti-symmetrization defines the biproduct  area(.,.), interpretable as the signed-area between two real-valued coordinate paths. We consider specific sets of binary, rooted trees known as Hall sets. These set have a complex combinatorial structure, which can be almost entirely circumvented by introducing the equivalent notion of Lazard sets. Using analytic results from dynamical systems and algebraic results from the theory of Lie algebras, we show that shuffle-polynomials in areas-of-areas on Hall trees generate the shuffle algebra.