  The aim of this chapter is to provide an adequate graph theoretic framework
for the description of periodic bifurcations which have recently been
discovered in descendant trees of finite p-groups. The graph theoretic concepts
of rooted in-trees with weighted vertices and edges perfectly admit an abstract
formulation of the group theoretic notions of successive extensions, nuclear
rank, multifurcation, and step size. Since all graphs in this chapter are
infinite and dense, we use methods of pattern recognition and independent
component analysis to reduce the complex structure to periodically repeating
finite patterns. The method of group cohomology yields subgraph isomorphisms
required for proving the periodicity of branches along mainlines. Finally the
mainlines are glued together with the aid of infinite limit groups whose finite
quotients form the vertices of mainlines. The skeleton of the infinite graph is
a countable union of infinite mainlines, connected by periodic bifurcations.
Each mainline is the backbone of a minimal subtree consisting of a periodically
repeating finite pattern of branches with bounded depth. A second periodicity
is caused by isomorphisms between all minimal subtrees which make up the
complete infinite graph. Only the members of the first minimal tree are
metabelian and the bifurcations, which were unknown up to now, open the long
desired door to non-metabelian extensions whose second derived quotients are
isomorphic to the metabelian groups. An application of this key result to
algebraic number theory solves the problem of p-class field towers of exact
length three.
