  The first part of this paper ( arXiv:1607.02114 ) introduced splitting trees,
those chronological trees admitting the self-similarity property where
individuals give birth, at constant rate, to iid copies of themselves. It also
established the intimate relationship between splitting trees and L\'evy
processes. The chronological trees involved were formalized as Totally Ordered
Measured (TOM) trees.
  The aim of this paper is to continue this line of research in two directions:
we first decompose locally compact TOM trees in terms of their prolific
skeleton (consisting of its infinite lines of descent). When applied to
splitting trees, this implies the construction of the supercritical ones (which
are locally compact) in terms of the subcritical ones (which are compact)
grafted onto a Yule tree (which corresponds to the prolific skeleton).
  As a second (related) direction, we study the genealogical tree associated to
our chronological construction. This is done through the technology of the
height process introduced by Duquesne and Le Gall. In particular we prove a
Ray-Knight type theorem which extends the one for (sub)critical L\'evy trees to
the supercritical case.
