  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
$\frac{1}{2}\log q$, there is a discrete subgroup $\Gamma$ acting without
inversion on a $(q+1)$-regular tree whose critical exponent is equal to
$\delta$. Explicit construction of edge-indexed graphs corresponding to a
quotient graph of groups are given.
