Algebraic two-level measure trees

With the algebraic trees, L\"ohr 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-$\sigma$-field, defining in this way an algebraic measure tree. The main result of L\"ohr 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.