Autoregressive Frontier Expansion: Growing Trees with Graph Machine Learning

Umer Gupta; Saku Peltonen; Martin Ritzert

cs.LGSep 29, 2026

Autoregressive Frontier Expansion: Growing Trees with Graph Machine Learning

Authors: Umer Gupta, Saku Peltonen, Martin Ritzert

Organizations: Independent Researcher London, United Kingdom · ETH Zurich Zurich, Switzerland · Leipzig University Leipzig, Germany

Abstract

Tree-like branching structures are common in nature, from botanical trees to neurons, blood vessels and respiratory trees. Their branching shape often reflects function, making structural modelling central to understanding how these systems work. Because acquiring real-world 3D data is often expensive or infeasible, realistic generative models are valuable for simulation and data augmentation. Existing morphology-specific models either constrain how topology is generated or rely on hand-tuned, mechanistic procedures. Generic 3D graph generators, by contrast, do not exploit or enforce the structure of trees. We propose Autoregressive Frontier Expansion, a generative framework that constructs trees through an iterative expansion process, simulating the biological growth of real trees. At each step, a flow-matching model parameterised by an SO(2)-equivariant GNN expands the frontier by predicting whether each active branch bifurcates or terminates. We evaluate our method on cortical neurons and botanical trees in unconditional, class-conditioned, and morphology-guided generation. Across both domains, the generated morphologies agree closely with the reference distributions and, in conditional experiments, with the specified targets.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of a generation step. Starting from a partial tree $T^{\ell}$ , we (i) expand the current frontier (a subset of leaves) by deciding for each frontier node whether it branches or terminates, yielding a tree $\widetilde{T}^{\ell+1}$ ; then (ii) localise by predicting 3D parent-relative offsets for the previous frontier nodes, producing $T^{\ell+1}$ . The process repeats until all frontier nodes terminate.

Figure 2: Stepwise generation of a botanical tree

Figure 3: Unconditional neuronal morphology generation. Top: Five representative generations from our model. Bottom: Distributions of eight morphology statistics for the training reference, SemlaFlow, MorphoGen, and our model. Dashed lines mark training-reference medians; grey values give the normalised

W_{1}

to the training reference (lower is better). Within-tree distributions weight each morphology equally, and horizontal ranges are determined from the training reference.

		Structure	Distributional agreement
Method	Params.	Valid tree (%) $\uparrow$	Morph. $\Delta\mathrm{MMD}^{2}$ $\downarrow$	TMD $\Delta\mathrm{MMD}^{2}$ $\downarrow$	Coverage $\uparrow$	Density $\to$ ref.	Mean norm. marginal $W_{1}$ $\downarrow$
Real–real	—	100.0	0.0000	0.0000	0.971	0.976	0.039
SemlaFlow	22.3M	76.6	0.2521	0.1038	0.223	0.155	0.389
MorphoGen	32.6M	100.0	0.8436	0.2077	0.006	0.018	1.515
Ours ( $F=64$ )	1.57M	100.0	0.0557	0.0146	0.808	0.827	0.175
Ours ( $F=128$ )	5.69M	100.0	0.0471	0.0108	0.805	0.862	0.163

Table 1: Unconditional neuronal morphology generation. Distributional metrics on

1{,}167

samples. The real–real row compares two disjoint training samples of the same size. Mean marginal

W_{1}

averages the six rotation-invariant statistics defined in Appendix F . Default for our method is

F=256

Figure 4: Class-conditioned neuronal morphology generation. (A) Class-wise morphology discrepancy for our model and SemlaFlow relative to the training trees. The values represent normalised 1-Wasserstein distance, averaged over five rotation-invariant statistics (lower is better). Whiskers show 95% bootstrap intervals, resampling the generated morphologies while holding the sampled training reference fixed. $n$ is the number of generated morphologies. (B) Two selected generations from our model per class. (C) Two selected training-set references per class.

Figure 5: TMD-conditioned generation. Top: Each row shows a conditioning target, one generated sample, their path-based persistence diagrams, and within-tree distributions of branch length, bifurcation angle and branch order. Bottom: 5 independent samples conditioned on the same target.

	Validity	Persistence diagrams		Within-tree distributions
Method	Valid tree (%) $\uparrow$	Path PD $W_{1}$ $\downarrow$	Radial-root PD $W_{1}$ $\downarrow$	Branch length $W_{1}$ ( $\mu$ m) $\downarrow$	Sibling angle $W_{1}$ ( ∘ ) $\downarrow$	Branch order $W_{1}$ $\downarrow$
Reference	100.0	5.914	6.453	16.15	8.34	1.190
SemlaFlow	63.2	3.158	2.453	18.01	10.61	0.668
Ours	100.0	2.939	2.803	13.38	9.50	0.548

Table 2: TMD-conditioned generation. Each method generates one morphology for each of 1,167 held-out targets. Entries in the persistence diagrams and within-tree distributions are means over target–generation pairs. The former compares normalised persistence diagrams, and the latter compares distributions of within-tree attributes including branch length, sibling angle, and branch order. The reference row pairs each target with a different held-out reference using a fixed-seed random derangement, providing a no-conditioning baseline.

Figure 6: Generated botanical trees across depth caps. Three representative samples are shown for each maximum topological depth

d_{\max}\in\{10,15,20\}

Regime	Method	Valid tree (%) $\uparrow$	Max. path length $\downarrow$	Branch length $\downarrow$	Bifurcation angle $\downarrow$	Partition asymmetry $\downarrow$	Mean $\downarrow$
Unconditional	SemlaFlow	3.9	1.280	0.245	0.297	0.822	0.661
Unconditional	Ours	100.0	0.443	0.167	0.177	0.157	0.236
TMD-conditioned	SemlaFlow	12.5	0.982	0.172	0.417	0.683	0.564
TMD-conditioned	Ours	100.0	0.165	0.087	0.238	0.076	0.141

Table 3: Botanical-tree generation. Population-level discrepancies on botanical trees restricted to depth 10 (full dataset). Morphology entries are tree-balanced 1-Wasserstein distances normalised by the reference standard deviation

W_{1}/\sigma_{\mathrm{ref}}

(lower is better). Mean averages the distances. Bold marks the better result per training regime.

Depth	Method	Max. path length $\downarrow$	Branch length $\downarrow$	Bifurcation angle $\downarrow$	Partition asymmetry $\downarrow$	Mean $\downarrow$	Time (s/tree)
D10	SemlaFlow	1.314	0.275	0.330	0.737	0.664	0.562
D10	Ours	0.542	0.178	0.197	0.148	0.266	0.192
D15	SemlaFlow	2.347	0.441	0.272	1.871	1.233	3.714
D15	Ours	0.639	0.227	0.200	0.139	0.301	0.390
D20	SemlaFlow	2.572	0.361	0.452	3.964	1.837	12.317
D20	Ours	0.702	0.250	0.198	0.366	0.379	0.617

Table 4: Unconditional botanical-tree generation across depth caps. Columns are

W_{1}/\sigma_{\mathrm{ref}}

discrepancies (lower is better), Mean averages the four discrepancies, and Time is seconds per tree. Bold marks the better result within each depth.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 7 — Figure 7: (Left) A tree with root $r$ . The nodes are embedded in $\mathbb{R}^{2}$ . Their coordinates induce Euclidean edge lengths, which define a distance-to-root filtration. (Middle) A barcode showing the resulting $H_{0}$ persistence under this filtration, with the elder rule deciding which branch survives each merge. The bars are annotated by ( $v_{d},v_{b}$ ), the death and birth nodes respectively. Note that the barcode itself is unlabelled. (Right) The corresponding persistence diagram. Points close to the diagonal represent short-lived features, while points further away represent more persistent features.

TMD channel	Assigned target $W_{1}$	Matched alternative $W_{1}$	Assigned/matched ratio $\downarrow$	Assigned closer (%) $\uparrow$
Path PD	2.916	5.543	0.526 $\pm$ 0.015	91.1 $\pm$ 0.8%
Radial-root PD	2.775	5.666	0.490 $\pm$ 0.015	90.8 $\pm$ 0.8%

Appendix

Table 5: Repeated sampling under TMD-conditioning. Persistence-diagram distances to the assigned target and to root-degree-matched alternatives across five generations per test neuron. “assigned closer” is the percentage of matched comparisons in which the assigned target has the lower distance. Variability between samples is much lower than to arbitrary neurons showing the effect of conditioning.

Figure 8: Error by cell type. Class-conditioned generation errors for our model (top) and SemlaFlow (bottom). Entries are class-wise $W_{1}$ normalised by the reference standard deviation (lower is better). Both models work well on the same cell types and ours clearly outperforms SemlaFlow, except for node count that is given to SemlaFlow as input and partition asymmetry where errors are generally low.

Figure 9: Unconditional generation for neurons. Reference morphologies, post-processed SemlaFlow generations, MorphoGen generations, and generations from our model are shown. Neurons are drawn root-centred and to scale within each method.

Figure 10: Class-conditioned neuron samples. Each row corresponds to one of the seven cortical pyramidal-cell classes; columns compare training examples, SemlaFlow generations, and generations from our model. The displayed samples are selected per source and class rather than treated as paired reconstructions.

Figure 11: Stochastic variation under TMD-conditioning. Each row shows one held-out conditioning target followed by five independent generations for that target. The generated samples preserve the target’s coarse dendritic organisation while varying in branch geometry and topology. Row C shows a more difficult condition for which the generations are visibly more compact than the target. B4 and E0 are unrealistic since they exhibit two apical dendrites instead of just one.

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Trees 3D Geometry 3D Generation Generative Models Norm

Open paper AskPDF arXiv

We introduce a novel mathematical framework for analyzing and generating complex tree-shaped 3D objects, such as botanical trees and plants, which deform both in their 3D geometry and branching structure. Unlike previous works, which either consider only the skeletal structure of tree-like objects or approximate their 3D geometry using branch thickness, the proposed framework accurately models both the 3D geometry of the tree branches and the way they are interconnected. In this paper, we first generalize the Square Root Normal Fields (SRNF) representation, originally proposed for the statistical analysis of genus-0 surfaces, to tree-shaped 3D objects. We then treat tree-shaped 3D objects as points on a novel Riemannian tree-shape space equipped with a novel Riemannian metric that measures the amount of surface bending and stretching, and structural changes one needs to apply to one 3D tree-shape to align it with another. This way, deformations become trajectories in this novel tree-shape space. We analyze the theoretical properties of this novel tree-shape space and the corresponding metric and develop algorithms for computing point-wise and branch-wise correspondences and geodesic paths between complex 3D trees. We finally show how to use these building blocks for (1) computing statistical summaries, \ie means and modes of variation, of collections of tree-shaped 3D objects, and (2) synthesizing novel tree-shaped 3D objects by sampling from probability distributions fitted to a population of tree-shaped 3D objects. We demonstrate the performance and utility of the proposed framework on real and synthetic plants and botanical trees and show that it significantly outperforms the state-of-the-art.

Abstract

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