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Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. We display a variety of examples of random combs and explain how they can be used in applications. In particular, we review some old and recent results regarding the genetic structure of the population when throwing neutral mutations on the skeleton of the tree.

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 tree grows higher than a given height, using a spinal decomposition of the tree, similar to the classical Bismut and Williams decompositions. 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. [ABSTRACT FROM AUTHOR]