  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-$\sigma$-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.
