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 Lévy trees. We use the Poissonian structure of the jumps of the increasing tree-valued process to describe its behavior at the first time the...

We use Poisson approximation techniques for sums of indicator random variables to derive explicit error bounds and central limit theorems for several functionals of random trees. In particular, we consider (i) the number of comparisons for successful and unsuccessful search in a binary search tree and (ii) internode distances in increasing trees. The Poisson approximation setting is shown to be...