Abstract. We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number delta between 0 and 1/2 log q, there is a discrete subgroup Gamma acting on a(q+1)-regular tree whose critical exponent is equal to delta . We explicitly construct an edge-indexed graph whose nite grouping has critical exponent delta . Moreover, we investigate the critical exponents of Schottky free discrete groups of rank 2 and give minimal polynomials of some examples.