TreeON: Tree-Centric Frameworks in Diverse Domains

• TreeON is a collection of frameworks that treat tree structures as primary entities, embedding them as core objects rather than auxiliary annotations.
• They employ diverse methodologies—from neural networks reconstructing 3D trees and combinatorial tests for phylogenetic networks to CAT(0) statistical analyses and adaptive search on tree domains.
• The systems extend to layered CRDT architectures that reconcile ordered and unordered trees, demonstrating trade-offs in metadata, convergence, and conflict resolution.

Searching arXiv for “TreeON” and directly related papers to ground the article in current indexed records. TreeON is not a single canonical system in the arXiv literature. The name has been used for several technically distinct frameworks organized around tree-structured objects: a neural framework for reconstructing individual-tree 3D point clouds from a single orthophoto and Digital Surface Model (DSM); a tree-oriented analysis framework in BHV phylogenetic treespace for Fréchet means, optimization, and regression; a paradigm for adaptive search structures over tree domains via Search Trees on Trees (STTs); a layered CRDT design for ordered and non-ordered trees; and, in one operational synthesis, a toolkit for recognizing tree-based phylogenetic networks via edgebasedness, generalized series-parallel structure, and Hamiltonicity-based sufficient criteria (Grammatikaki et al., 11 Mar 2026, Skwerer, 2014, Berendsohn et al., 2020, 1201.00931, Fischer et al., 2018, Martin et al., 2012).

1. Scope and conceptual orientation

Across these usages, TreeON consistently denotes a framework in which the tree is not merely an output format but the primary structural object. In the geospatial reconstruction setting, the target object is an individual tree represented as a compact colored 3D point cloud reconstructed from sparse top-down observations. In phylogenetic-network recognition, the central object is an unrooted phylogenetic network and the problem is to decide or certify treebasedness. In BHV treespace, trees are points in a CAT(0) metric space and support statistical inference through geodesics and Fréchet means. In STTs, the domain itself is a tree and search costs are defined over rooted decompositions of that domain. In CRDTs, trees are replicated data objects whose rootedness, uniqueness of paths, and optional sibling order must be preserved under concurrency (Grammatikaki et al., 11 Mar 2026, Fischer et al., 2018, Skwerer, 2014, Berendsohn et al., 2020, Martin et al., 2012).

A plausible implication is that the shared name reflects a recurring methodological commitment: tree structure is treated as an inductive bias, a constraint set, or a state invariant rather than as a secondary annotation. The technical consequences differ sharply by domain. Some TreeON variants are predictive and differentiable, some are combinatorial and certificate-oriented, some are geometric and optimization-theoretic, and some are distributed and convergence-theoretic.

2. Neural TreeON for 3D tree point-cloud reconstruction

In "TreeON: Reconstructing 3D Tree Point Clouds from Orthophotos and Heightmaps" (Grammatikaki et al., 11 Mar 2026), TreeON is a neural framework for reconstructing detailed 3D colored point clouds of individual trees using only a single RGB orthophoto patch and its aligned DSM patch. The geospatial setup uses a top-down orthographic camera aligned along the $Z$ axis and centered at the tree’s bounding box origin. DSMs are generated by taking the $Z$-buffer depth from the orthographic camera, converting it to elevation, subtracting local ground elevation, and scaling to the unit range $[0,1]$. For network use, DSM pixels are back-projected into 3D, yielding a point cloud that preserves the canopy’s vertical profile and sharp height discontinuities. Query points for occupancy prediction are sampled within a unit cube $[0,1]^3$ in normalized coordinates.

The architecture is explicitly bimodal. The orthophoto is encoded by a ResNet-18 backbone pre-trained on ImageNet, followed by pooled intermediate features and a projection head with batch normalization and LeakyReLU, producing a 1024-D image embedding that is $\ell_2$-normalized. The DSM is encoded by a PointNet-style network with 1D convolutions of 64, 128, and 1024 channels, again followed by batch normalization and LeakyReLU; max pooling yields a 1024-D latent vector that is also $\ell_2$-normalized. Concatenation produces a 2048-D fused latent vector. Query points are Fourier-encoded with 25 frequency bands to a 153-D representation, so the decoder input effectively has size 2201 per query. The occupancy decoder is a deep MLP with six fully connected layers, each using Layer Normalization, dropout, LeakyReLU, and residual connections. Two heads are produced: a scalar occupancy logit and a 3-channel RGB color vector. A lightweight refinement module of three residual MLP layers adds a corrective delta to occupancy logits to sharpen surface boundaries and recover fine-scale detail.

Supervision combines point-level occupancy, color consistency, and differentiable projection constraints. Occupancy labels are assigned to query points by nearest-neighbor distance to a reference point cloud obtained by Poisson-disk sampling of the mesh, with binary cross-entropy denoted $\mathcal{L}_{\text{occ}}$. Color uses an $L_2$ loss $\mathcal{L}_{\text{col}}$. Shadows are rendered differentiably with a SoftRas-like projection under the known sun direction and compared to ground-truth shadow masks with LPIPS, yielding $\mathcal{L}_{\text{shadow}}$. Silhouettes are rendered from five canonical viewpoints—$Z$0, $Z$1, $Z$2, $Z$3, and top view—and compared with LPIPS to obtain $Z$4. The reported weights are $Z$5, $Z$6, $Z$7, and $Z$8.

The training set is synthetic. TreeON uses 3,000 procedural trees across 17 species—Beech, Oak, Birch, Hornbeam, Alder, Aspen, Poplar, Ash, Linden, Elm, Maple, Field Maple, Plane Tree, Fir, Austrian Pine, Scots Pine, and Stone Pine—with proportions reflecting alpine forestry at approximately $Z$9 conifers and $[0,1]$0 deciduous. Grove-generated meshes provide species-specific geometry and inline vertex colors. Ground-truth point clouds contain approximately $[0,1]$1 uniformly distributed colored points per tree. Orthophotos are rendered with Blender’s Cycles using a directional light with azimuth sampled from $[0,1]$2, elevation from $[0,1]$3, and sun angular diameter $[0,1]$4. Terrain patches come from 12 DEMs covering $[0,1]$5 km. DSM resolution is set to 1 m. The split is 2,500 trees for training with an $[0,1]$6 train/validation split and 500 trees held out for testing, ablations, and baseline comparisons.

Training uses Adam with initial learning rate $[0,1]$7, ReduceLROnPlateau with halving after 20 epochs of stalled validation loss, batch size 16, and 700 epochs on a single NVIDIA GeForce RTX 3070 (8 GB), requiring approximately 24 hours. Inference time is approximately 0.3 seconds per tree. The output is the top-$[0,1]$8 set of points by occupancy probability after sigmoid, optionally upsampled by voxelization and resampling with nearest-neighbor color propagation; shaded-sphere rendering uses PCA-estimated normals.

Quantitatively, the mixed DSM+Orthophoto configuration with BCE+Shadow+Silhouette supervision reports $[0,1]$9, $[0,1]^3$0, $[0,1]^3$1, $[0,1]^3$2, and $[0,1]^3$3. The paper also reports retrained baseline comparisons on the TreeON dataset: Ours gives $[0,1]^3$4, $[0,1]^3$5, $[0,1]^3$6, and $[0,1]^3$7, compared with Diffusion PC at $[0,1]^3$8, $[0,1]^3$9, $\ell_2$0, $\ell_2$1, 3DTopia-XL at $\ell_2$2, $\ell_2$3, $\ell_2$4, $\ell_2$5, and OpenLRM at $\ell_2$6, $\ell_2$7, $\ell_2$8, $\ell_2$9. The reported interpretation is not taxonomic reconstruction fidelity but visual plausibility, structural plausibility, efficiency, and automation. The paper states explicitly that TreeON prioritizes visual plausibility over species-accurate morphology and identifies failure cases involving rare species or seasonal appearances, dead trees, absent shadows, and misleading DSM shapes.

3. TreeON for tree-based phylogenetic network recognition

A separate operationalization frames TreeON as a recognition and certification pipeline for tree-based phylogenetic networks, based on the graph-theoretic results summarized from "Classes of treebased networks" (Fischer et al., 2018). In the unrooted setting, an unrooted phylogenetic network on a finite taxon set $\ell_2$0 is a connected, simple graph $\ell_2$1 with $\ell_2$2, no vertex of degree 2, and leaf set bijectively labeled by $\ell_2$3. The network is treebased if there exists a spanning tree $\ell_2$4 with $\ell_2$5 such that the leaf set of $\ell_2$6 equals $\ell_2$7; such a tree is called a support tree. The associated decision problem is NP-complete in the unrooted case.

The central tractable subclass is edgebasedness. Leaf shrinking allows four operations: deleting a leaf and its incident edge, suppressing a degree-2 vertex, deleting one copy of a parallel edge, and deleting a loop. A connected graph is edgebased iff the leaf-shrinking process reduces it to $\ell_2$8. The main structural theorem states that a connected graph is GSP iff it is loopless and edgebased, and a connected graph is GSP iff every block is an SP graph. Because GSP graphs can be recognized in linear time, edgebasedness can be checked in linear time. Two implementation routes are given: a block-decomposition pipeline using Hopcroft–Tarjan plus per-block SP recognition such as Valdes–Tarjan–Lawler, or a direct leaf-shrink simulation with dynamic degree counts and adjacency lists. Both yield complexity $\ell_2$9.

For a proper phylogenetic network, edgebasedness is a sufficient condition for treebasedness. The proof strategy is constructive: every nontrivial SP block admits a spanning tree whose leaves are the degree-2 vertices of the block, and the union of these spanning trees with all cut edges yields a global support tree whose leaf set is exactly the taxa. The paper further gives a blobwise characterization: a proper unrooted phylogenetic network with at least two leaves is edgebased iff every nontrivial blob is edgebased. This enables modular certification.

The framework also defines LCUT, LS, and LCON reductions and several Hamiltonicity-based sufficient conditions. A network is H-connected if it is proper, has at least two leaves, and $\mathcal{L}_{\text{occ}}$0 is Hamilton connected. Sufficient criteria for treebasedness include the existence of leaves $\mathcal{L}_{\text{occ}}$1 with adjacent attachment points $\mathcal{L}_{\text{occ}}$2 and a $\mathcal{L}_{\text{occ}}$3–$\mathcal{L}_{\text{occ}}$4 path visiting all inner vertices, Hamilton connectedness of $\mathcal{L}_{\text{occ}}$5, and Hamiltonian graphs in $\mathcal{L}_{\text{occ}}$6 satisfying specified edge-creation conditions. A corollary states that if some $\mathcal{L}_{\text{occ}}$7 is a 10-tough chordal graph, then $\mathcal{L}_{\text{occ}}$8 is treebased. Another theorem states that if $\mathcal{L}_{\text{occ}}$9 is binary and chordal, then it is edgebased and hence treebased.

The section also clarifies a common point of confusion: edgebasedness is strictly stronger than treebasedness. The paper exhibits a network that is treebased but not edgebased, so failure of the edgebased test does not imply non-treebasedness. This suggests a certification-oriented TreeON workflow: first use the linear-time edgebasedness/GSP test, preferably blobwise; if that fails, apply LCUT- and LCON-based Hamiltonicity heuristics or family-specific sufficient conditions rather than attempt exact recognition in the NP-complete general case.

4. TreeON as tree-oriented statistical analysis in BHV treespace

In the treespace-oriented formulation synthesized from "Tree Oriented Data Analysis" (Skwerer, 2014), TreeON denotes an analytical framework built on BHV phylogenetic treespace $L_2$0. Fixing $L_2$1 labeled leaves indexed by $L_2$2, each phylogenetic tree is encoded by compatible splits and positive edge lengths. BHV treespace is constructed by gluing Euclidean nonnegative orthants $L_2$3, one per compatible split set, with maximal orthants of dimension $L_2$4. The space is complete, globally non-positively curved, and CAT(0). Consequently, for any two trees $L_2$5, there is a unique global geodesic $L_2$6.

The geodesic structure is controlled by a support sequence of incompatible edge groups $L_2$7 and a common-edge set $L_2$8, subject to properties (P1) compatibility, (P2) ratio monotonicity, and (P3) no further improving partitions. With $L_2$9 and $\mathcal{L}_{\text{col}}$0, the BHV distance is

$\mathcal{L}_{\text{col}}$1

The Owen–Provan algorithm computes geodesics with worst-case complexity $\mathcal{L}_{\text{col}}$2 per pair.

This geometry supports Fréchet inference. For data trees $\mathcal{L}_{\text{col}}$3 and weights $\mathcal{L}_{\text{col}}$4, the Fréchet function is

$\mathcal{L}_{\text{col}}$5

and the Fréchet mean is the unique minimizer because squared distance is geodesically convex in CAT(0) spaces and $\mathcal{L}_{\text{col}}$6 is strictly geodesically convex, continuous, and proper. The strict uniqueness of the mean remains valid even when the minimizer lies on a lower-dimensional face of treespace and corresponds to a degenerate topology such as a star tree. This removes a misconception imported from non-CAT(0) settings: topological degeneracy does not imply mean nonuniqueness.

Algorithmically, the framework combines global split proximal point algorithms with local damped Newton refinement. Inductive means and cyclic SPPA provide globally convergent iterates in CAT(0) spaces. Within a fixed orthant, the restricted gradient and Hessian of $\mathcal{L}_{\text{col}}$7 are available in closed form, and line search uses Armijo and Wolfe conditions while enforcing edge-length nonnegativity. Vistal and multivistal cells capture regions in which geodesic supports remain constant; support updates along line segments are handled by detecting intersections with (P2) and (P3) hyperplanes and using parametric maximum-flow logic on incompatibility bipartite graphs. The stated combined system consists of outer SPPA iterations followed by local Newton steps, edge deletion below tolerance $\mathcal{L}_{\text{col}}$8, and stopping criteria based on gradient components over active and inactive edges.

The framework also develops asymptotics. For a probability measure $\mathcal{L}_{\text{col}}$9 on $\mathcal{L}_{\text{shadow}}$0 with finite first moment, the population Fréchet mean $\mathcal{L}_{\text{shadow}}$1 minimizes

$\mathcal{L}_{\text{shadow}}$2

When $\mathcal{L}_{\text{shadow}}$3 lies on a lower-dimensional face, the behavior of normal directional derivatives partitions the measure into sticky, partly sticky, and non-sticky regimes. The sticky law of large numbers states that, with probability 1, there exists a finite random $\mathcal{L}_{\text{shadow}}$4 such that for all $\mathcal{L}_{\text{shadow}}$5, the topology of the sample Fréchet mean stabilizes to that of the population mean in the sticky case, and stabilizes in the specified partial sense in the partly sticky case.

A further component is nonparametric regression of trees against a scalar predictor. With weights

$\mathcal{L}_{\text{shadow}}$6

the conditional Fréchet mean is

$\mathcal{L}_{\text{shadow}}$7

The motivating application is MRA-derived human brain artery data from 109 nominally healthy subjects with 64 landmarks per hemisphere, yielding 129 leaves in $\mathcal{L}_{\text{shadow}}$8. For the full 128-landmark representation, the sample Fréchet mean is reported as a star tree or nearly star, reflecting strong topological heterogeneity and geodesics that dip sharply toward the origin. Reduced-landmark analyses show declining interior-edge ratios beyond approximately 24 landmarks. With 17 landmarks and ages 20–80, Gaussian-kernel smoothing at bandwidths 1–6 years indicates decreasing total interior edge length with age, a positive correlation between number of interior edges and total interior length, and collapse to a single representative topology for $\mathcal{L}_{\text{shadow}}$9–6.

5. TreeON as adaptive search on tree-structured domains

In the STT formulation connected to "Splay trees on trees" (Berendsohn et al., 2020), TreeON denotes the paradigm of operating search structures over tree-structured domains. Let $Z$00 be an undirected, unrooted tree. A search tree on $Z$01 is a rooted tree $Z$02 with $Z$03 defined recursively: removing the root of $Z$04 from $Z$05 yields connected components, and each child subtree of the root is itself an STT on one such component. The edges of $Z$06 need not be edges of $Z$07; the key invariant is that the child subtrees partition the components of $Z$08. Binary search trees arise as the special case in which $Z$09 is a path.

Search cost has both static and dynamic forms. Under a distribution $Z$10, the expected static cost of an STT $Z$11 is

$Z$12

In the dynamic model, each access begins at the current root and may perform pointer moves and rotations, each costing 1, followed by an additional search cost 1. A rotation swaps a node with its parent while reassigning children so that the STT component invariant is preserved. Rotation distance is the minimum number of such rotations needed to transform one STT into another on the same domain tree.

The paper’s first major contribution is a PTAS for static optimization. It defines $Z$13-cut STTs, where each subtree $Z$14 has boundary size at most $Z$15, and shows that the family of $Z$16-admissible subproblems has cardinality $Z$17. For any integer $Z$18, one can compute an STT $Z$19 in time $Z$20 satisfying

$Z$21

The construction first transforms any STT into a $Z$22-cut STT with controlled depth inflation and then runs dynamic programming over connected subgraphs indexed by cuts of size at most $Z$23. This restores a Knuth-like optimization program that is impossible over arbitrary exponentially many connected subtrees.

The second major contribution is an adaptive self-adjusting structure, SplayTT, on Steiner-closed STTs, which are exactly the 2-cut STTs. The paper proves linear rotation distance inside the $Z$24-cut family: any two $Z$25-cut STTs on the same $Z$26-node tree can be transformed into one another in at most

$Z$27

rotations, while keeping all intermediate trees $Z$28-cut. This is significant because unrestricted STTs can have rotation diameter $Z$29.

SplayTT generalizes zig, zig-zig, and zig-zag to the STT setting. A new complication absent in BSTs is branching in the convex hull of the search path inside the underlying tree $Z$30. The algorithm therefore uses a two-phase strategy: first, a branch-elimination phase splays branching nodes until the search path becomes nonbranching in the required sense; second, a root-splay phase splays the accessed node to the root. The tree remains Steiner-closed throughout. The resulting static optimality theorem states that for a sequence $Z$31 of $Z$32 searches,

$Z$33

and the additive $Z$34 term disappears if every node is accessed at least once. The analysis uses a combinatorial potential indexed by depths in an optimal reference Steiner-closed STT rather than the entropy-based access lemma familiar from BST splay analysis.

The paper presents these results as evidence for an extended dynamic optimality conjecture on tree domains. That conjecture remains open. Accordingly, TreeON in this sense is best viewed as a partial transfer of the BST optimality program to arbitrary tree domains rather than a completed theory.

6. TreeON as a layered CRDT architecture for ordered and non-ordered trees

In "Abstract unordered and ordered trees CRDT" (Martin et al., 2012), TreeON expands to "Trees Ordered and Non-ordered" and denotes a layered approach to tree CRDTs. The design separates three concerns: a base convergent set representation, tree-specific correction layers, and an independent positioning layer for sibling order. The underlying unordered tree is a rooted directed tree $Z$35 satisfying acyclicity, connectivity from the root, and the single-parent property. An equivalent word-tree view encodes the tree as a prefix-closed subset of $Z$36.

The base representation uses set CRDTs. The paper describes graph-tree, edge-tree, and word-tree variants instantiated with G-Set, 2P-Set, LWW-Set, PN-Set, and OR-Set, yielding names such as GG-Tree, 2G-Tree, LG-Tree, CG-Tree, OG-Tree, and their edge- and word-based analogues. In the graph-tree model, $Z$37 adds $Z$38 to $Z$39 and $Z$40 to $Z$41; remove operations delete a subtree or node according to the semantics of the underlying set CRDT. In the word-tree model, adding a child is adding a path extension, while removing a subtree is removing a prefix-closed set of paths.

Naive lookups may violate rootedness, uniqueness of paths, or acyclicity under concurrency, so TreeON inserts correction layers. The first layer is a connection policy repairing rootedness. Four policies are given. skip drops orphan components entirely by traversing from the root. reappear recreates deleted ancestor paths using tombstones so that orphan components reconnect at their original place. root attaches orphan components directly under the root. compact attaches them under the nearest existing ancestor(s) that historically reached the deleted ancestors, using retained historical connectivity. The second layer is a mapping policy that converts the rooted graph into a tree. several preserves all simple acyclic paths from the root, allowing logical duplication of a node in multiple locations. one chooses a spanning arborescence, with tie-breaking variants such as newer, higher, or shortest. zero deletes any node with indegree greater than 1 together with its subtree. The paper notes that several is monotonic but may blow up combinatorially, whereas one and some connection policies are non-monotonic because nodes may move when new ancestry becomes visible.

Ordering is handled independently through a positioning layer. For unique position identifiers, TreeON can adopt sequence-CRDT schemes such as Treedoc, Logoot, WOOT, WOOTO, or RGA. The paper also introduces WOOTR (Recursive-WOOT), a non-unique positioning scheme whose elements are triples $Z$42 with recursively defined predecessor and successor references between sentinels $Z$43 and $Z$44. This allows concurrent same-place insertions to converge without duplicating the element. Ordering can be attached either to nodes or to edges, depending on the structural representation.

Correctness is compositional. The state space of the underlying CvRDT is a join-semilattice $Z$45 with commutativity, associativity, and idempotence, and final convergence follows from monotone merges $Z$46. TreeON then defines a deterministic lookup pipeline from merged set state to corrected rooted graph, mapped tree, and positioned ordered tree. Because the correction and positioning functions are deterministic and depend only on convergent CRDT state, all replicas converge to the same lookup. The paper states rootedness and tree-property lemmas for the correction policies and an eventual-consistency theorem for the resulting ordered or unordered tree.

The main trade-offs are explicit. Add-wins versus remove-wins behavior depends on the underlying set CRDT. reappear and compact require tombstones or historical summaries. one-newer aligns naturally with LWW metadata, while one-higher aligns with PN-Set or OR-Set counters or tags. skip + one-shortest gives efficient and predictable behavior, reappear + one-newer preserves original structure at higher metadata cost, root + zero aggressively isolates conflicts, and several preserves concurrency at the cost of potentially large lookup size.

7. Cross-cutting structure, misconceptions, and technical limits

Taken together, these TreeON variants show that the name functions as a recurrent label for systems in which trees are central state spaces, output manifolds, or invariants rather than a single interoperable software stack. A plausible implication is that TreeON is best understood as a family resemblance across domains rather than a unified method.

Several recurrent misconceptions are addressed directly by the underlying works. In phylogenetic recognition, treebasedness is not equivalent to edgebasedness; the latter is only a stronger sufficient condition (Fischer et al., 2018). In BHV analysis, degeneracy of the Fréchet mean on a lower-dimensional face does not violate uniqueness, because strict geodesic convexity is preserved in CAT(0) treespace (Skwerer, 2014). In STTs, the transfer of BST machinery is only partial: linear rotation-distance and static optimality hold in restricted families such as Steiner-closed or $Z$47-cut STTs, while unrestricted dynamic optimality remains conjectural (Berendsohn et al., 2020). In the geospatial reconstruction setting, visually coherent 3D trees should not be interpreted as biologically faithful morphology; the paper explicitly states that the model prioritizes visual plausibility over species-accurate structure (Grammatikaki et al., 11 Mar 2026). In CRDTs, eventual consistency does not imply a unique semantic notion of conflict resolution, because TreeON deliberately exposes multiple independent policy choices for rootedness, mapping, and sibling order (Martin et al., 2012).

The technical limits are equally domain-specific. Geospatial TreeON depends on the synthetic training distribution and lacks systematic real-world paired benchmarks. Phylogenetic TreeON inherits NP-completeness for exact recognition and therefore emphasizes tractable sufficient conditions. BHV TreeON must handle nonsmoothness and ill-conditioned Hessians near orthant boundaries. STT TreeON faces the unresolved problem of proving dynamic optimality beyond static optimality. CRDT TreeON trades metadata overhead, monotonicity, and data preservation against one another depending on the chosen policies.

What remains common is the insistence that tree structure is operationally consequential. Whether the goal is reconstructing a tree crown from orthophotos, certifying support trees inside phylogenetic networks, averaging anatomical trees in CAT(0) geometry, adapting search structures over nonpath domains, or preserving replicated hierarchical state, TreeON denotes a program of making tree structure explicit, algorithmically tractable, and central to the semantics of the system (Grammatikaki et al., 11 Mar 2026, Fischer et al., 2018, Skwerer, 2014, Berendsohn et al., 2020, Martin et al., 2012).