  Based on the work of Abercrombie, Barnea and Shalev gave an explicit formula
for the Hausdorff dimension of a group acting on a rooted tree. We focus here
on the binary tree T. Abert and Virag showed that there exist finitely
generated (but not necessarily level-transitive) subgroups of AutT of arbitrary
dimension in [0,1].
  In this article we explicitly compute the Hausdorff dimension of the
level-transitive spinal groups. We then show examples of 3-generated spinal
groups which have transcendental Hausdroff dimension, and exhibit a
construction of 2-generated groups whose Hausdorff dimension is 1.
