UFO Trees: Robotics & Dynamic Data Structures

• UFO Trees are dual-use systems representing both a mechanization-friendly training method for sweet cherries and a dynamic data structure for parallel batch operations.
• In horticulture, UFO Trees use standardized architectures and CNN-guided semantic skeletonization to facilitate precise robotic pruning and yield optimization.
• In algorithmic design, UFO Trees support dynamic queries with O(min{log n, D}) operation times and efficient batch updates, ensuring scalable performance in dynamic forests.

UFO Trees are a term used in both horticultural robotics and dynamic data structures, denoting: 1) Upright Fruiting Offshoot (UFO) trees—a specialized, mechanization-oriented cultivation form for sweet cherry; and 2) Unbounded Fan-Out trees—a provably efficient parallel batch-dynamic data structure for dynamic forests. Both contexts are actively studied and referenced as benchmarks and frameworks in their respective domains (You et al., 2021, Man et al., 15 Jan 2026).

1. Upright Fruiting Offshoot Trees: Definition and Botanical Significance

Upright Fruiting Offshoot (UFO) trees constitute a training system for sweet cherry (Prunus avium) characterized by a standardized architecture optimized for high-yield and mechanization. The standard UFO form involves:

• A main trunk from which two horizontal supports (cordons) emerge at approximately 0.6 m above ground,
• Seven to nine vertical leader branches originating from the supports, spaced 0.15–0.45 m apart, and rising to about 3 m,
• Lateral side-branches along leaders, which bear the fruiting spurs.

This design facilitates mechanical pruning and harvesting, while dormant pruning—removal of old or underproductive side-branches—enhances bud renewal and yield, although manual execution is labor‐intensive and requires horticultural expertise. Standardized topology (trunk → support → leader → side-branch) lends itself to codified, automation-ready semantic rules (You et al., 2021).

2. Skeletonization of UFO Trees for Robotic Pruning

Reconstructing an accurate and semantically labeled tree skeleton is a key enabler for automation in orchard management. The 1D skeleton $S = (N, E, L)$ models the tree as a set of superpoints (nodes $N$), directed edges $E$ between nodes, and edge labels $L(e)$ in $$\{\text{Trunk}, \text{Support}, \text{Leader}, \text{SideBranch}, \text{None}\}$$.

The pipeline for semantics-guided skeletonization comprises:

• Superpoint Extraction: The raw point cloud $P \subset \mathbb{R}^3$ is overlaid with spheres ($r_{\text{super}} = 0.10$ m); sphere centers are retained as superpoints.
• Dense Edge Proposal & CNN Scoring: Candidate edges connect superpoints within $2r_{\text{super}}$. Each edge proposal is classified by a convolutional neural network using a rasterized representation of local geometry, yielding a confidence score $\mathrm{Conf}(e) \in [0,1]$.
• Tip and Trunk Identification: Tip (vertical leader ends) and trunk base nodes are identified using a minimum spanning forest and spatial heuristics; trunk base selection is manual.
• Population-Based Skeleton Growth: $K=500$ skeleton candidates are grown by expanding toward tip superpoints under skeleton reward function $\mathrm{SkelScore}(S)$ that incorporates edge confidence, turn penalty, and label-aligned growth penalty. Only edge–label additions consistent with UFO-specific topology (e.g., precisely two supports emerge from the trunk) are allowed.
• Label-Rich Post-Processing: Leader–side branch junctions are refined via detection of major directional changes. Low-confidence edges are pruned.

Semantic labeling enforces biologically valid pruning actions and topology (e.g., no supports growing from side-branches) (You et al., 2021).

3. Evaluation, Dataset, and Empirical Results in Robotic Contexts

The semantic skeletonization algorithm has been systematically validated:

• Dataset: 82 raw 3D scans of UFO trees (Washington State orchard, December 2019), of which 43 filtered clouds underpin the annotated benchmark; 14 clouds are reserved for CNN training/validation (15,000 edge examples).
• Testing Protocol: 29 trees, each downsampled three times to 50,000 points, generate 87 skeletonization trials.
• Metrics: Median global edit ratio (fraction of edge additions, deletions, or label changes required to match human ground truth) is approximately 30%, translating to about 70% semantic skeleton accuracy. Per-label edit errors indicate highest reliability for trunk detection (0%), moderate for leaders (21%), but more error in support (42%) and side-branch (50%) assignment, mainly due to superpoint ambiguity at complex junctions.
• Data Accessibility: The full scan corpus is publicly available for reproducibility and further research.

This methodology provides an initial but critical step toward robust tree modeling for autonomous pruning systems (You et al., 2021).

4. UFO Trees as a Dynamic Data Structure: Context and Construction

In algorithmic graph theory, UFO trees (Unbounded Fan-Out trees) are a dynamic forest maintenance data structure optimized for both rich query support and efficient parallel batch updates (Man et al., 15 Jan 2026).

Key features include:

• Operations Supported: link($u,v$), cut($u,v$), connectivity($u,v$); path and subtree aggregates, LCA, diameter, center, nearest-marked vertex, etc.
• Core Design: UFO trees perform bottom-up tree contraction via cluster merges. High-degree clusters (degree $\geq 3$) merge in a single round with all degree-one neighbors ("unbounded fan-out"), in contrast to topology trees/rake-compress trees, which require ternarization.
• Cluster Maintenance: Each cluster stores pointers to its parent, an adjacency array and hash-set, an optional aggregate, and update markers.
• Parallel Batch-Dynamics: Edge updates are grouped and processed level-wise in parallel; high-degree roots are merged immediately, remaining roots matched and merged via list ranking and maximal matching, propagating changed clusters up recursively.
• Space Usage: $O(n)$ total clusters in an $n$-vertex forest, achieved without ternarization.

Cluster contraction reduces the number of clusters by at least a factor of $5/6$ per round, yielding $O(\log n)$ contraction rounds; diameter-based contraction further accelerates processing on low-diameter inputs (Man et al., 15 Jan 2026).

5. Theoretical Guarantees and Algorithmic Performance

UFO trees admit several provable performance and functionality bounds:

• Height and Space: Contraction tree height is $O(\log n)$, and at most $\lceil D/2 \rceil$ for diameter $D$. Total clusters $O(n)$, total space $O(n)$.
• Operation Costs: Each sequential link, cut, connectivity, or path/subtree query takes $O(\min\{\log n, D\})$ worst-case time. For batches of $k$ updates, expected work is $O(\min\{k\log(1+n/k), kD\})$, with parallel span $O(\log n \log k)$.
• Query Functionality: Full support for connectivity, commutative-associative path aggregates, invertible subtree queries, LCA, diameter, center, median, and nearest-marked searches at the above cost levels. For non-invertible aggregates, $O(\log n)$ per query is required (with lower bound via reduction from comparison-sorting).

These results render UFO trees the first known structure to unify all major dynamic-tree queries with both efficient parallelism and without space-inefficient ternarization (Man et al., 15 Jan 2026).

6. Practical Implementation and Empirical Validation

UFO trees have been implemented in C++ leveraging ParlayLib for scalable multi-core parallelism. Design choices include:

• Fixed-size arrays and hash-sets for adjacency.
• Minimal cluster metadata (pointer, neighbor list, flags).
• Efficient batch update kernels: ParlayLib semisort for adjacency, custom parallel maximal matching via list ranking, streamlined memory allocation, and ancestor deletion avoidance.
• No ternarization required at any step.

Empirical results encompass:

• Sequential Performance: UFO trees are within a small constant (about $3\times$ slower) of link-cut trees and significantly faster (5–20$\times$) than dynamic-tree alternatives including RC, topology, and ETT on full query suites; on low-diameter inputs, UFO trees can outperform link-cut trees due to the diameter-parameterized cost.
• Parallel Scaling: On a 96-core system, UFO trees achieve 70$\times$ speedups on batch updates, within 1.5$\times$ of the specialized parallel Euler-tour tree (which lacks broad query support).
• Scalability: Able to handle billion-size trees owing to $O(n)$ space usage, where ETT and RC run out of memory.

This suggests UFO trees are both a performant and practical foundation for dynamic graph algorithms in large-scale settings (Man et al., 15 Jan 2026).

7. Practical Implications and Future Directions

In robotics and horticulture, semantically-labeled skeletons of UFO cherry trees enable automation of dormant pruning, transforming complex, skill-dependent manual tasks into programmable manipulations (e.g., automatic side-branch removal to specification). Limitations presently include side-branch detection accuracy and manual trunk base selection. Future directions include integrating adaptive probabilistic priors, improved superpoint strategies, extension to other trained fruit tree architectures, and branch-diameter integration for finer pruning granularity (You et al., 2021).

For dynamic trees in algorithmic theory, UFO trees provide the first data structure capable of supporting the full array of dynamic-tree queries, with guaranteed $O(\min\{\log n, D\})$ operation times, work-efficient parallel batch updates, linear space, and avoidance of ternarization. Prospective advancements involve further optimization for batch-processing in distributed settings, broadening the spectrum of supported aggregate queries, and applying UFO trees as subroutines in more complex graph algorithms (Man et al., 15 Jan 2026).