https://hal-cnrs.archives-ouvertes.fr/hal-03740750Nussbaumer, JosuéJosuéNussbaumerLAMA - Laboratoire Analyse et Mathématiques Appliquées - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - CNRS - Centre National de la Recherche Scientifique - Université Gustave EiffelTran, Viet-ChiViet-ChiTranLAMA - Laboratoire Analyse et Mathématiques Appliquées - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - CNRS - Centre National de la Recherche Scientifique - Université Gustave EiffelWinter, AnitaAnitaWinterUniversität Duisburg-Essen [Essen]Algebraic two-level measure treesHAL CCSD2022Algebraic treehierarchical structurenested treebranch point mapcontinuum treetriangulation of the circleconvergence of treestwo-level sample shape convergenceGromov-weak convergence[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]Tran, Viet ChiLaboratoires d'excellence - Models and algorithms: from the discrete to the continuous - - Bézout2010 - ANR-10-LABX-0058 - LABX - VALID - 2022-07-29 19:49:502022-08-03 04:00:412022-08-01 15:03:05enPreprints, Working Papers, ...application/pdf1With the algebraic trees, Löhr and Winter (2021) introduced a generalization of the notion of graph-theoretic trees to account for potentially uncountable structures. The tree structure is given by the map which assigns to each triple of points their branch point. No edge length or distance is considered. One can equip a tree with a natural topology and a probability measure on the Borel-σ-field, defining in this way an algebraic measure tree. The main result of Löhr and Winter is to provide with the sample shape convergence a compact topology on the space of binary algebraic measure trees. This was proved by encoding the latter with triangulations of the circle. In the present paper, we extend this result to a two level setup. Motivated by the study of hierarchical systems with two levels in biology, such as host-parasite populations, we equip algebraic trees with a probability measure on the set of probability measures. To show the compactness of the space of binary algebraic two-level measure trees, we enrich the encoding of these trees by triangulations of the circle, by adding a two-level measure on the circle line. As an application, we define the two-level algebraic Kingman tree, that is the random algebraic two-level measure tree obtained from the nested Kingman coalescent.