Axially Symmetric Phylogenetic Trees

• Axially symmetric phylogenetic trees (ASPTs) are trees with 2n leaves obtained from an axially symmetric subdivision of a regular 2n-gon with mirror-labeling, ensuring a unique reflective automorphism.
• They provide a concrete combinatorial model that identifies the tropicalization of type C cluster varieties with a polyhedral fan structured by symmetric weight spaces.
• ASPTs bridge tropical geometry and Catalan combinatorics by yielding subfans combinatorially equivalent to associahedra and cyclohedra, which capture symmetric edge contractions.

Axially symmetric phylogenetic trees are phylogenetic trees with $2n$ leaves, labeled by $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$, that arise from an axially symmetric subdivision of a regular $2n$-gon together with an axially symmetric labeling of its sides relative to a fixed longest diagonal. In the type $C$ setting, they provide an explicit combinatorial model for tropical geometry: the tropicalization of a type $C$ cluster variety is identified with the space of such trees, and the corresponding signed tropicalizations appear as subfans dual to associahedra or cyclohedra (Makhlin, 16 Feb 2026).

1. Definition through symmetric polygonal subdivisions

Let $n\geq 3$, and let

$N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$

A phylogenetic tree in this setting is a tree with $2n$ leaves, equipped with a bijective labeling $\lambda$ of the leaves by the elements of $N$, and with no degree-$N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$0 vertices. The construction used for axially symmetric phylogenetic trees begins with a regular $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$1-gon $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$2, a subdivision $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$3 by pairwise non-crossing diagonals together with the sides of $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$4, and a labeling $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$5 of the sides by $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$6. The dual graph of this subdivision produces a phylogenetic tree $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$7 (Makhlin, 16 Feb 2026).

The symmetry is imposed relative to a fixed longest diagonal $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$8 of $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$9. A subdivision is axially symmetric with respect to $2n$0 when it is closed under reflection across $2n$1. A labeling is axially symmetric when sides symmetric with respect to $2n$2 are labeled $2n$3 and $2n$4 for each $2n$5. An axially symmetric phylogenetic tree (ASPT) is then defined to be a tree of the form $2n$6 for which both the subdivision and the labeling are axially symmetric.

A central structural property is that each ASPT admits a unique involutive automorphism $2n$7, called the symmetry, that swaps $2n$8. This makes the object more specific than a generic phylogenetic tree with automorphisms: the symmetry is not merely an abstract graph automorphism, but one induced by axial reflection in the polygonal model. A common misconception is therefore to identify ASPTs with all symmetric phylogenetic trees. In the type $2n$9 construction, the class is narrower and is tied to a specific reflection-compatible combinatorial realization.

2. Weighted ASPTs and the fan $C$0

The weighted version of the theory uses axially symmetric weighted phylogenetic trees (ASWPTs). Such an object is an ASPT together with a weight function $C$1 on the edges that is positive on non-leaf edges and $C$2-invariant, meaning

$C$3

For an ASWPT $C$4, the distance function $C$5 is the sum of the edge weights along the path from $C$6 to $C$7. This distance is symmetric under the involution.

To record distances up to the imposed symmetry, the relevant index set is

$C$8

The data of all possible distance matrices for ASPTs lives in $C$9, and the weight assignment is written

$C$0

For each ASPT $C$1, the set of possible weights defines a relatively open cone

$C$2

The space of ASPTs is the polyhedral fan

$C$3

This fan has an explicit global structure: $C$4 is a pure fan of dimension $C$5, with lineality $C$6, and its facets correspond to contractions along symmetric edge orbits (Makhlin, 16 Feb 2026). Thus the space of axially symmetric trees is not only a set of combinatorial types; it is a polyhedral moduli space organized by the degeneration of symmetric edge data.

3. Identification with the tropicalization of a type $C$7 cluster variety

Let $C$8 be the type $C$9 cluster variety, specifically the type $n\geq 3$0 cluster algebra realized as an affine variety. The affine realization is given by a surjection from

$n\geq 3$1

onto $n\geq 3$2, with kernel $n\geq 3$3. Its tropicalization is

$n\geq 3$4

the fan in $n\geq 3$5 consisting of weight vectors for which the initial ideal $n\geq 3$6 is monomial-free (Makhlin, 16 Feb 2026).

The main theorem identifies these two structures: $n\geq 3$7 Accordingly, the tropical cluster variety of type $n\geq 3$8 coincides with the space of axially symmetric phylogenetic trees. Modulo lineality, the cones arising from ASPT weights and the cones of the tropicalization are canonically identified. The combinatorics of symmetric phylogenetic trees therefore gives an explicit description of the tropicalization.

A corollary extends the statement to any full rank geometric type $n\geq 3$9 cluster algebra: the same identification persists modulo the lineality space induced by scalings among variables. The paper presents this as a direct generalization of the type $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$0 case associated with the Grassmannian and the work of Speyer–Sturmfels. In that sense, ASPTs play for type $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$1 the role that ordinary phylogenetic tree spaces play in the classical type $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$2 tropical Grassmannian picture.

4. Signed tropicalizations, dihedral orderings, and Catalan duality

The signed tropicalization framework begins with a sign pattern

$N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$3

Let $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$4 denote the automorphism multiplying variables by their assigned signs. The signed tropicalization $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$5 is defined as the positive tropicalization of $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$6. The theory yields a precise enumeration: there are

$N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$7

distinct signed tropicalizations. Of these,

$N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$8

are combinatorially equivalent, modulo lineality, to the dual fan of an associahedron, and

$N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}.$9

are combinatorially equivalent to the dual fan of a cyclohedron (Makhlin, 16 Feb 2026).

The combinatorial mechanism uses dihedral orderings of the $2n$0 sides of $2n$1 up to rotation and reflection. Two special classes are distinguished. Axially symmetric dihedral orderings (ASDOs) are compatible with the axial symmetry, while centrally symmetric dihedral orderings (CSDOs) encode center symmetry. For each ASDO, the compatible ASPTs determine a subfan combinatorially equivalent, modulo lineality, to the dual face fan of an associahedron. For each CSDO, the compatible centrally symmetric phylogenetic trees determine a subfan combinatorially equivalent to the dual face fan of a cyclohedron.

This is formalized by

$2n$2

where $2n$3 is an ASDO or a CSDO. Geometrically, axially symmetric subdivisions of $2n$4 are in bijection with subdivisions of an $2n$5-gon, while centrally symmetric subdivisions correspond to faces of the cyclohedron. The result places ASPTs inside Catalan combinatorics: the relevant subfans are not arbitrary slices of $2n$6, but are organized by associahedral and cyclohedral duality.

5. Relation to other symmetry notions in phylogenetics

A distinct but related line of work studies symmetries of rooted, non-embedded, leaf-labelled binary trees under relabelling by permutations. In that setting, $2n$7 acts on trees with $2n$8 leaves, and for $2n$9, the quantity $\lambda$0 counts the trees fixed by $\lambda$1. If the cycle-type of $\lambda$2 is the partition $\lambda$3, then $\lambda$4 is nonzero if and only if $\lambda$5 is a binary partition, and in that case

$\lambda$6

(Fusy, 2016).

A special case occurs when $\lambda$7 has a single cycle and hence $\lambda$8: then $\lambda$9, so there is exactly one tree, the perfectly balanced, completely symmetric binary tree. The same formula feeds into the enumeration of tanglegrams and tangled chains: $N$0 The paper also gives a combinatorial proof of the fixed-tree formula and describes a simplified random sampler for tangled chains.

These results concern a different symmetry problem from the ASPT theory. In the permutation-fixed setting, symmetry is formulated through invariance under relabelling actions on rooted binary trees. In the ASPT setting, symmetry is encoded by reflection across a longest diagonal of a regular $N$1-gon together with the mirror pairing $N$2. The two frameworks are therefore complementary rather than interchangeable.

6. Mathematical significance and adjacent developments

The ASPT construction has several stated mathematical consequences. It generalizes the Speyer–Sturmfels type $N$3 correspondence between tropicalizations and phylogenetic trees to type $N$4; it links the tropical geometry of cluster varieties to concrete moduli of symmetric metric trees; and it connects the resulting fan structures to Catalan objects such as associahedra and cyclohedra (Makhlin, 16 Feb 2026). The fan $N$5 and its subfans provide a polyhedral realization of the tropical cluster variety and its signed tropicalizations, while the explicit cone structure gives a face description in terms of symmetric tree combinatorics.

The scope of the construction should be stated precisely. ASPTs are not the space of all phylogenetic trees with $N$6 leaves, nor are signed tropicalizations arbitrary decompositions of the tropicalization. The ambient space is $N$7, where $N$8 records pair indices up to the involution; the trees have $N$9 leaves; and the relevant symmetries are axial or central in the polygonal model. The distinction matters because the associahedral and cyclohedral subfans arise from compatibility with ASDOs and CSDOs, not from a generic symmetry condition.

In adjacent work on computational phylogenetics, tropical methods have also been used algorithmically rather than purely combinatorially. A separate paper introduces Tropical Axial Attention, which replaces vanilla softmax dot-product attention with max-plus operators and leverages the well known isomorphic relationship between the space of all phylogenetic trees with $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$00 species and tropical Grassmannian (Teska et al., 12 May 2026). That work is not a theory of ASPTs, but it indicates a broader movement in which tropical geometry, tree-space constraints, and symmetry-aware constructions are used both for structural descriptions of moduli spaces and for phylogenetic inference. A plausible implication is that explicit fan models such as $N=\{1,\dots,n,\overline{1},\dots,\overline{n}\}$01 may become increasingly relevant wherever tree-metric consistency and symmetry constraints need to be represented in a concrete polyhedral form.