  In this third paper, we revisit the question to which extent the properties
of the trees associated to the tilings $\{p,4\}$ of the hyperbolic plane are
still true if we consider a finitely generated tree by the same rules but
rooted at a black node? What happens if, considering the same distinction
between black and white nodes but changing the place of the black son in the
rules. What happens if we change the representation of the numbers by another
set of digits?
  We tackle all of these questions in the paper. The present paper is an
extension of the previous papers arXiv:1904.12135 and arXiv:1907.04677.
