  We improve the previously best known lower and upper bounds on the number n_g
of numerical semigroups of genus g. Starting from a known recursive description
of the tree T of numerical semigroups, we analyze some of its properties and
use them to construct approximations of T by generating trees whose nodes are
labeled by certain parameters of the semigroups. We then translate the
succession rules of these trees into functional equations for the generating
functions that enumerate their nodes, and solve these equations to obtain the
bounds. Some of our bounds involve the Fibonacci numbers, and the others are
expressed as generating functions.
  We also give upper bounds on the number of numerical semigroups having an
infinite number of descendants in T.
