Exit times for an increasing L\'evy tree-valued process

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.