Spaces of algebraic measure trees
and triangulations of the circle
In this paper we investigate algebraic trees which can be considered as (continuum) metric trees in which the metric distances are ignored and in which therefore the focus lies on the tree structure. We give an axiomatic definition of such trees, which we call algebraic trees, using a branch point map and show that any order separable algebraic tree can be represented by a metric tree. We further consider algebraic measure trees which are algebraic trees additionally equipped with a sampling (probability) measure. This measure gives rise to the branch point distribution which turns out to be the length measure of an intrinsic choice of such a metric tree representation.
We will provide a notion of convergence of algebraic measure trees which resembles the idea of the Gromov-weak topology which itself is defined through weak convergence of sample distance matrices.
Binary algebraic (measure) trees are of particular interest because they often arise in praxis, and also due to their close connection to triangulations of the circle. We use this connection to show that in the subspace of binary algebraic measure trees, weak convergence of sample shapes, sample subtree masses and sample distance matrices are all equivalent and define a compact, metrizable topology. Furthermore, the coding by triangulations is a continuous, surjective operation in this topology.
2 Algebraic trees
- 2.1 The branch point map
- 2.2 Morphisms of algebraic trees
- 2.3 Algebraic trees as topological spaces
- 2.4 Metric tree representations of algebraic trees
2.5 Tree homomorphisms versus homeomorphisms
3 The space of algebraic measure trees
4 Triangulations of the circle
- 4.1 The space of sub-triangulations of the circle
- 4.2 Coding binary measure trees with (sub-)triangulations of the circle
- 4 Triangulations of the circle
- 3 The space of algebraic measure trees
- 2 Algebraic trees
In the recent years the construction and investigation of scaling limits of tree-valued Markov chains became more and more of interest. This started with the continuum analogs of the Aldous-Broder-algorithm for sampling a uniform spanning tree from the complete graph ([EPW06]) and of the tree-valued subtree-prune and regraft Markov chain used in the reconstruction of phylogenetic trees ([EW06]). It continued with the construction of evolving genealogies of infinite size populations in population genetics ([GPW13, DGP12, Pio10, GSW16]) and in population dynamics ([Glo12, KW]). Moreover, continuum analogues of pruning procedures were constructed ([ADV10, AD12, LVW15, HWb, HWa]).
In order to provide a unified set-up which includes graph-theoretical trees on the one hand and continuum trees on the other, it became by now a classic approach to encode trees as metric (measure) spaces or bi-measure -trees and to equip the space of all metric (measure) trees with the Gromov-Hausdorff ([Gro99]), Gromov-weak ([Fuk87, Fuk87, GPW09, Löh13]) or Gromov-Hausdorff-weak ([Vil09, ADH13, ALW17]) topology and the space of all bi-measure -trees with the leaf-sampling weak-vague topology ([LVW15]). All these approaches have in common that they rely on encoding trees as metric spaces.
With the present paper we want to present a set-up which rather than focusing on pairwise metric distances brings the attention to the tree structure. We thereby want to respond to the observation that sometimes the intrinsic graph distance does not seem to be the right notion of distance. It might, for example, behave too wildly to allow the suitably rescaled family of tree-valued Markov chains to be tight, and consequently let the metric approach fail.
The goal of this paper is to overcome the metric issue by focusing on the tree structure only. We will introduce algebraic measure trees with an axiomatic approach, but it turns out that they can (under a separability constraint) be viewed as metric trees where one has “forgotten” the metric. To this end, we introduce an intrinsic metric which comes from the branch point distribution (Definition 3.5).
Definition 1.1 (-trees).
A metric space is an -tree iff it satisfies the following:
satisfies the so-called -point condition, i.e., for all ,
is a connected metric space.
Notice that any metric space satisfying (RT1) and (RT2) admits a branch point map , i.e., for all there exists a unique point such that
where here for
While condition (RT1) is crucial for trees as it reflects the fact that there is only one possible shape for the subtree spanned by four points (as shown in Figure 1), the assumption of connectedness can be relaxed. In [ALW17], the notion of a metric tree was introduced as a metric space which can be embedded isometrically into an -tree such that it contains all branch points , , as defined by (1.2). To exclude non-tree graphs satisfying the -point condition (see, for example, Figure 1), we have to require the property of containing the branch points explicitly.
Definition 1.2 (Metric trees).
A metric space is a metric tree if the following two conditions hold:
satisfies the -point condition (1.1).
admits all branch points, i.e., for all there exists a (necessarily unique) such that
Our main goal is to forget the metric while keeping the tree structure encoded by the branch point map. To axiomatize the latter, notice that for metric trees the branch point map satisfies the following obvious properties:
The map is symmetric.
The map satisfies the -point condition, i.e., for all
The map satisfies the -point condition, i.e., for all
The map satisfies the -point condition, i.e., for all ,
We therefore define the following:
Definition 1.3 (Algebraic tree).An algebraic tree consists of a set and a branch point map satisfying (BPM1)–(BPM4).
We want to consider algebraic trees as topological spaces where the topology is generated as follows: For each point , we define an equivalence relation on such that for all , iff . For , we denote by
the equivalence class w.r.t. which contains . Clearly, any equivalence class corresponds to a subtree rooted at (but not containing) in the embedding -tree. We consider the topology generated by sets of the form (1.9) with and denote by the corresponding Borel -algebra.
Our first main result (Theorem 1) relates metric trees with algebraic trees. On the one hand, if is a metric tree, then it is clear that together with the map from (MT2) yields an algebraic tree. On the other hand, we shall show that every order separable algebraic tree, i.e. a separable tree with at most countably many edges, is induced by a metric tree in this way. More concretely, we will show that if is a measure on which is finite and non-zero on non-degenerate intervals, i.e., on sets of the form
for , , then a metric representation of is given by
We next want to equip an algebraic tree with a sampling (probability) measure on . An algebraic measure tree consists of an algebraic tree and a probability measure on . Two algebraic measure trees and are equivalent if there are , and a bijection such that the following holds.
, and .
is measure preserving, and for all .
(Compare with Definition 3.2.)
Denote by the space of all equivalence classes of order separable algebraic measure trees. We shall equip with a notion of convergence based on the Gromov-weak topology. For that purpose, we introduce a particular metric representation of an algebraic measure tree. As metric representations are far from being unique, we will consider the intrinsic metric which comes from the branch point distribution, i.e., the image measure of under the branch point map . We declare that
and refer to this convergence as branch point distribution distance Gromov-weak convergence, or shortly, bpdd-Gromov-weak convergence.
A particular subclass of interest is the space of binary algebraic measure trees. Similar to encoding compact -trees by a continuous excursion on the unit interval, binary algebraic trees can be encoded by triangulations of the circle (see Figure 3).
Such an encoding was introduced originally by David Aldous in [Ald94a, Ald94b], and there has since then been an increasing amount of research in the random tree community using this approach (e.g. [CLG11, BS15, CK15]). Also more general -angulations and dissections have been considered which allow for encoding not necessarily binary trees ([CHK15]).
Aldous’s originally defines a triangulation of the circle as a closed subset of the disc the complement of which is a disjoint union of open triangles with vertices on the circle (see [Ald94b, Definition 1]). We modify Aldous’s definition in two respects. First, we add a condition which excludes non-tree graphs (such a condition is missing in Aldous’s definition) and under which triangulations of the circle are precisely the Hausdorff-metric limits of triangulations of -gons. Second, we extend the definitions to so-called sub-triangulation of the circle (triangulations of a subset of the circle) which allow for encoding not only the algebraic tree but the measure on it in such a way that it is allowed to have point masses on leaves. In fact, any triangulation of the whole circle encodes a binary tree with a non-atomic measure which is relevant in the case of Aldous’s CRT. We then show that sub-triangulations of the circle can indeed be used to encode binary algebraic measure trees with point-masses restricted to the leaves. Furthermore, we show that – similar to the case of coding compact -trees by continuous excursions – the coding map that associates to a sub-triangulation of the circle the corresponding algebraic measure tree is surjective and continuous when the set of sub-triangulations is equipped with the Hausdorff metric topology and the set of binary algebraic measure trees with bpdd-Gromov-weak topology (Theorem 2).
We also analyze the space of binary algebraic measure trees with point-masses restricted to the leaves in more detail. Our third main result (Theorem 3) states that this space is a compact, metrizable space under the bpdd-Gromov-weak topology. We also give two more notions on convergence. One is based on weak convergence of the tree-shapes spanned by a finite sample. The other on weak convergence of the tensor of subtree-masses read off the algebraic measure subtree spanned by a finite sample. It turns out that all three notions of convergence are equivalent on this subspace.
It is obvious that our results can be extended easily to trees with a bound other than three on the degree. However, in order to consider unbounded (or infinite) degrees additional care has to be taken. The reasons is that such subspaces of algebraic measure trees can not any longer expected to be compact.
Close relatives of algebraic measure trees have been recently studied in Forman [For]. The equivalence of bpdd-Gromov-weak topology and weak convergence of sample tree-shapes is related to the space of didentritic systems introduced recently by Evans, Grübel, and Wakolbinger in [EGW17]. Didentritic systems can be considered as ordered binary algebraic measure trees, and the space of didentritic systems is equipped with a kind of sample shape convergence.
Outline. The rest of the paper is organized as follows. In Section 2, we introduce our concept of algebraic trees by formalising the branch point map as a tertiary operation on the tree. We show that under a separability constraint algebraic trees can be seen as subtrees of metric trees, where the metric structure has been “forgotten” (Theorem 1).
In Section 3, we introduce the space of (equivalence classes of) order separable algebraic measure trees, and equip it with the Gromov-weak topology with respect to the metric associated with the branch point distribution.
In Section 4, we give a definition and characterisation of triangulations of the circle. We also formalize the notion of the algebraic (measure) tree associated with a given triangulation of the circle. This correspondence has often been pointed out in the literature but has never been made precise (except for discrete, graph-theoretic trees where it is more or less obvious). We show that the resulting coding map (which associates a triangulation of the circle with a tree) is well-defined and surjective onto the space of binary algebraic measure trees with non-atomic measure (Theorem 2).
In Section 5, we restrict ourselves to the subspace of binary, order separable, algebraic measure trees, and introduce two other, natural notions of convergence. We will use the construction of the coding map from Section 4 to show that on the subspace of binary algebraic measure trees, all three notions of convergence define the same topology (Theorem 3). This topology turns our space of binary algebraic measure trees into a compact, metrizable space. Furthermore, we show that the coding map is continuous if the space of triangulations is equipped with the Hausdorff metric topology, and the space of trees with the bpdd-Gromov-weak topology.
2 Algebraic trees
In this section we introduce algebraic trees. In Subsection 2.1 we formalize the “tree-structure” common to both graph-theoretic trees and metric trees by a function that maps every triplet of points in the tree to the corresponding branch point. We show that the set of defining properties is rich enough to obtain known concepts such as leaves, branch points, degree, edges, intervals, subtrees spanned by a set, discrete and continuum trees, etc. In Subsection 2.2 we introduce the notion of structure preserving morphisms. In Subsection 2.3 we equip algebraic trees with a canonical Hausdorff topology. We also characterize compactness and a concept we call order separability, which is closely related to second countability of the topology. Finally, in Subsection 2.4, we show that any order separable algebraic tree is induced by a metric tree (which is not true without order separability), and establish the condition under which this metric tree can be chosen to be a compact -tree.
2.1 The branch point map
In this subsection we introduce algebraic trees. Recall from Definition 1.2 the definition of a metric tree, and the properties (BPM1)–(BPM4) of the map which sends a triplet of points in a metric tree to its branch point.
Definition 2.1 (algebraic trees).
An algebraic tree consists of a set and a branch point map satisfying (BPM1)–(BPM4).
The following useful property reflects the fact that any four points in an algebraic tree can be associated with a shape as illustrated in Figure 1 above.
Lemma 2.2 (a consequence of (BPM4)).
Let be an algebraic tree. Then for all the following hold:
If , then .
If , then .
Let with , and .
(i) Condition (BPM4) implies that
Thus , or . The second case is the claim. In the first case, we apply Condition (BPM4) once more to find that
so that the claim also holds in this case.
(ii) Condition (BPM3) implies that
and similarly also . Now part (i) with replaced by yields as claimed. ∎
We have seen that the four axiomatizing properties of the branch point map are necessary. In many respects they are also sufficient to capture the tree structure. For example, in analogy to (1.3) we can define for each the interval by
Lemma 2.3 (properties of intervals).
Let be an algebraic tree. Then the following hold:
If are such that and , then .
If , then
If , then
For all ,
(i) Let with and . Then by Condition (BPM4),
If then by Lemma 2.2(i). Thus , and the claim holds also in this case.
Equivalently, . This proves the inclusion . The other inclusion follows from (i).
(iii) Notice first that it follows immediately from (i) that the union on the right hand side is disjoint. We claim that
Indeed, let , i.e. . Then by (BPM4) applied to ,
which implies that (if ) or (if ) or (if ). Second, we claim that for all ,
To see this, recall from (ii) that . As by (i), we have . The corresponding inclusion for is shown in the same way, and we have proven Equation (2.7).
(iv) This follows immediately from (ii). ∎
We say that with is an edge of if and only if there is “nothing in between”, i.e. , and denote by
the set of edges. The following example explains that there is no need to distinguish between finite algebraic trees and graph-theoretical trees, and the definitions of edges are consistent.
Example 2.4 (finite algebraic trees correspond to graph-theoretic trees).
Finite algebraic trees are in one to one correspondence with finite (undirected) graph-theoretic trees. Let be a graph-theoretic tree with vertex set and edge set . Then corresponds to the algebraic tree with defined as the unique vertex that is on the (graph-theoretic) path between any two of . Conversely, if is an algebraic tree with finite, then corresponds to the graph-theoretic tree with . Obviously, . ∎
For a graph-theoretic tree , we can allow the vertex set to be countably infinite, and still obtain a corresponding algebraic tree as in the previous example. Note, however, that countable algebraic trees do not necessarily correspond to graph-theoretic trees. Indeed, it is possible that is countably infinite and . This can be seen by taking in the following example, which shows that every totally ordered space naturally corresponds to an algebraic tree.
Example 2.5 (totally ordered spaces as algebraic trees).
For a totally ordered space , define whenever , (). Then it is trivial to check that is an algebraic tree and the interval coincides with the order interval . ∎
Conversely, given an algebraic tree and any fixed point (often referred to as root), we can define a partial order by letting for ,
Lemma 2.6 (algebraic trees as semi-lattices).
Let be an algebraic tree, and . Then is a partially ordered set, and a meet semi-lattice with infimum
Furthermore, is a total order on for all .
Let with and . That is, and which implies that , and proves that is antisymmetric. As , which proves that is reflexive. Finally, to show transitivity, let with and . That is and , which implies that by Lemma 2.3(i). Equivalently, which proves the transience, and thus that is a partial order.
Fix . For totality on , let , i.e., and . Applying Condition (BPM4) to we find that one of the following three cases must occur: (which implies that , or equivalently, ), (which implies that , or equivalently, ), or (which implies that ). ∎
Let be an algebraic tree, and . If , then .
Let and . That is, . We need to show that .
The partial orders allow us to define a notion of completeness of algebraic trees.
Definition 2.8 (directed order completeness).
Let be an algebraic tree. We call (directed) order complete if for all the supremum of every totally ordered, non-empty subset exists in the partially ordered set .
Obviously, in an order complete algebraic tree, infima of totally ordered sets exists, because they are either if the set is empty or a non-empty supremum w.r.t. a different root. This notion of completeness allows us to define the analogs of complete -trees.
Definition 2.9 (algebraic continuum tree).
We call an algebraic tree algebraic continuum tree if the following two conditions hold:
is order complete.
2.2 Morphisms of algebraic trees
Like any decent algebraic structure (or in fact mathematical structure), algebraic trees come with a notion of structure-preserving morphisms.
Definition 2.10 (morphisms).
Let and be algebraic trees. A map is called a tree homomorphism (from into ) if for all ,
We refer to a bijective tree homomorphism as tree isomorphism.
As we have seen that the tree structure can be expressed also in terms of intervals or partial orders rather than the branch point map, and we obtain the following equivalences.
Lemma 2.11 (equivalent definitions).
Let and be algebraic trees, and . Then the following are equivalent:
is a tree homomorphism.
For all , is an order preserving map from to .
For all , .
The image of an algebraic tree under a homomorphism is a subtree in the following sense.
Definition 2.12 (subtree).
Let be an algebraic tree, and . is called a subtree (of ) if
We refer to as the algebraic subtree generated by .
Obviously, a subtree of , implicitly equipped with the restriction of to , is an algebraic tree in its own right. Furthermore, the following lemma is easy to check.
Lemma 2.13 (tree homomorphisms).
Let and be two algebraic trees, and a homomorphism. Then the image is a subtree of . If is injective, is a tree homomorphism from into .
In particular, if is another algebraic tree, and is a homomorphism form to , then is a homomorphism from to .
2.3 Algebraic trees as topological spaces
In contrast to metric trees, there is a priori no topology defined on a given algebraic tree. In this section, we therefore equip algebraic trees with a canonical topology.
For each , we introduce a (component) relation by letting if and only if , where . Let for each
be the equivalence class of containing , and note that is a subtree for all , and whenever . We refer to as the component of containing . Now and in the following, we equip with the topology
generated by the set of components, i.e. with the coarsest topology such that all components are open sets. We call the component topology of .
Example 2.14 (on totally ordered trees, is the order topology).
If is a totally ordered space, and the corresponding algebraic tree as in Example 2.5, then coincides with the order topology (i.e. the one generated by sets of the form and for ). ∎
Example 2.15 (intervals are closed sets).
Let be an algebraic tree, and . Then
This means that is closed in the component topology . ∎
Let be an algebraic tree. Then is continuous w.r.t. the component topology .
By definition of , it is sufficient to show that for any , , the set is open in . By definition of and the property shown in Lemma 2.3, if and only if (at least) two of are in . Because is open, the same is true for in the product topology. Hence is a union of open set and thus open. ∎
Next, we show that is a Hausdorff topology and characterize compactness of algebraic trees in this topology.
Lemma 2.17 ( is Hausdorff).
Let be an algebraic tree. Then the component topology defined in (2.22) is a Hausdorff topology on .
To show that is Hausdorff, let be distinct. If , then and are clearly disjoint neighbourhoods of and , respectively. Assume that this is not the case, and choose . Then . Furthermore, , and hence . Thus and are disjoint neighbourhoods of and , respectively. Hence is Hausdorff. ∎
Proposition 2.18 (characterizing compactness).
Let be an algebraic tree with component topology . Then is compact if and only if is directed order complete.
“only if”. Assume first that is not order complete. Then we can choose and such that is totally ordered w.r.t. but does not have a supremum in . For , let and . Then and are open sets. We claim that is an open cover of . Indeed, if , then, because has no supremum, there is with , hence . Otherwise, if , there is with . Thus is a cover of .
has no finite sub-cover, because if were such a finite sub-cover, then would cover . This, however, would imply that would be a supremum of , contradicting our assumption. Hence is not compact.
“if”. Assume that is order complete. Consider a cover of with components, i.e. . By the Alexander subbase theorem, for compactness of , it is sufficient to show that has a finite sub-cover.
To this end, fix an element and consider the set . By Hausdorff’s maximal chain theorem (or Zorn’s lemma), there is a maximal chain in the partially ordered set . For every , we have , and thus there is such that . We claim that implies . Indeed, and hence which is equivalent to . Therefore, exists in by directed order completeness of . Because is a cover, there is with , hence for some . Because and is a maximal chain, . Hence there is with . We claim that . Indeed, let . Then . Using and , we obtain , and hence . Thus as claimed, and is the desired sub-cover. ∎
It turns out that the following separability condition, which we will call order separability, is crucial for us.
Proposition 2.19 (order separability).
Let be an algebraic tree with component topology . Then the following are equivalent:
There exists a countable set such that for all with ,
The topological space is second countable (i.e. has a countable base), and is countable.
The topological space is separable, and is countable.
satisfies (2.24). Indeed, is countable by assumption. Moreover, let . Then two cases are possible: either . In this case, , which implies that . Or . In this case, as is open by definition of , there is . Let . Then , and either , or the three components , , are distinct. In the second case, we can choose and to see that . In any case, .