  In this paper we elaborate on the structure of the semigroup tree and the
regularities on the number of descendants of each node observed earlier. These
regularites admit two different types of behavior and in this work we
investigate which of the two types takes place in particular for well-known
classes of semigroups. Also we study the question of what kind of chains appear
in the tree and characterize the properties (like being (in)finite) thereof. We
conclude with some thoughts that show how this study of the semigroup tree may
help in solving the conjecture of Fibonacci-like behavior of the number of
semigroups with given genus.
