In this paper, we study the Markoff-Hurwitz equations x0 + ...+x 2 n = ax0 · · ·xn. The variety V defined by this equation admits a group of automorphisms A ∼= Z/2∗· · ·∗Z/2 (an n+1 fold free product). For a solution P on this variety, we consider the number NP (t) of points Q in the A-orbit of P with logarithmic height h(Q) less than t. We show that if a is rational, and P is a non-trivial rational solution to this equation, then the limit