  The goal of this work is to decompose random populations with a genealogy in
subfamilies of a given degree of kinship and to obtain a notion of infinitely
divisible genealogies. We model the genealogical structure of a population by
(equivalence classes of) ultrametric measure spaces (um-spaces) as elements of
the Polish space U which we recall. In order to then analyze the family
structure in this coding we introduce an algebraic structure on um-spaces (a
consistent collection of semigroups). This allows us to obtain a path of
collections of subfamilies of fixed kinship h (described as ultrametric measure
spaces), for every depth h as a measurable functional of the genealogy. Random
elements in the semigroup are studied, in particular infinitely divisible
random variables. Here we define infinite divisibility of random genealogies as
the property that the h-tops can be represented as concatenation of independent
identically distributed h-forests for every h and obtain a Levy-Khintchine
representation of this object and a corresponding representation via a
concatenation of points of a Poisson point process of h-forests. Finally the
case of discrete and marked um-spaces is treated allowing to apply the results
to both the individual based and most important spatial populations. The
results have various applications. In particular the case of the genealogical
(U-valued) Feller diffusion and genealogical (U V -valued) super random walk is
treated based on the present work in [DG18b] and [GRG]. In the part II of this
paper we go in a different direction and refine the study in the case of
continuum branching populations, give a refined analysis of the Laplace
functional and give a representation in terms of a Cox process on h-trees,
rather than forests.
