  The notion of tree network has sparked renewed interest in recent years,
particularly in computer science and biology (neural network). However, this
notion is usually interpreted in an extremely restrictive way: essentially
linked to data processing, today tree networks are hybrid network topologies in
which star networks are generally interconnected via bus networks. These
networks are, most often, hierarchical and regular, and each of their nodes can
have an arbitrary number of child nodes. At the outset, however, the notion of
tree network, introduced in 1936 by Belgian physicist Julien Pacotte, was quite
different: more general and, at the same time, more constrained, it should also
serve an ambitious objective: the reconstruction of mathematics from concrete
empirical structures. Usually poorly commented on and poorly understood
(especially by philosophers), it had no real posterity. In this article, we
first try to clarify this notion of "tree network" in the sense of Julien
Pacotte, which makes it possible to eliminate the bad interpretations to which
this notion has given rise. To this end, we use the language and concepts of
graph theory and formalize the main properties of these networks which,
contrary to popular belief, are not, in general, trees. In a second part, we
then try to follow and explain, step by step, how Pacotte intended, using
concepts borrowed from projective geometry, to reconstruct all of mathematics
from such a network.
