  We give an explicit construction of the increasing tree-valued process
introduced by Abraham and Delmas using a random point process of trees and a
grafting procedure. This random point process will be used in companion papers
to study record processes on L\'evy trees. We use the Poissonian structure of
the jumps of the increasing tree-valued process to describe its behavior at the
first time the tree grows higher than a given height. We also give the joint
distribution of this exit time and the ascension time which corresponds to the
first infinite jump of the tree-valued process.
