Spaces of algebraic measure trees and triangulations of the circle

Abstract

In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures.For that purpose we focus on the tree structure given by the branch point map which assigns to each triple of points their branch point. We give an axiomatic definition of algebraic trees, define a natural topology, and equip them with a probability measure on the Borel-σ-field.Under an order-separability condition, algebraic (measure) trees can be considered as tree structure equivalence classes of metric (measure) trees (i.e. subtrees of R-trees). Using Gromov-weak convergence (i.e. sample distance convergence) of the particular representatives given by the metric arising from the distribution of branch points, we define a metrizable topology on the space of equivalence classes of algebraic measure trees.In many applications binary trees are of particular interest. We introduce on that subspace with the sample shape and the sample subtree mass convergence two additional, natural topologies. Relying on the connection to triangulations of the circle, we show that all three topologies are actually the same, and the space of binary algebraic measure trees is compact. To this end, we provide a formal definition of triangulations of the circle, and show that the coding map which sends a triangulation to an algebraic measure tree is a continuous surjection onto the subspace of binary algebraic non-atomic measure trees. Contents 1. Introduction 2 2. Algebraic trees 8 2.1. The branch point map 8 2.2. Morphisms of algebraic trees 11 2.3. Algebraic trees as topological spaces 12 2.4. Metric tree representations of algebraic trees 15 2.5. Tree homomorphisms versus homeomorphisms 20 3. The space of algebraic measure trees 21 4. Triangulations of the circle 27 4.1. The space of sub-triangulations of the circle 28 4.2. Coding binary measure trees with (sub-)triangulations of the circle 31 5. The subspace of binary algebraic measure trees 34 5.1. Convergence in distribution of sampled tree shapes 34 5.2. Convergence in distribution of sampled subtree masses 37 5.3. Equivalence and compactness of topologies 41 6. Example: sampling consistent families 42 Appendix A. A uniform Glivenko-Cantelli theorem 44

Exchangeable hierarchies and mass-structure of weighted real trees

“…A parallel effort by Löhr, Mytnick, and Winter [24] has proven existence of a continuum analogue to Aldous's Markov chain on a space of "algebraic measure trees." The algebraic measure trees, introduced in [25] concurrently with and independently of the present work, serve a similar purpose to IP trees in representing mass-structure, but the authors take a more algebraic approach. There is an obvious distinction in that the objects of the present work are all rooted trees, whereas algebraic trees are unrooted.…”