Generalized Fruit Tree Models

• Generalized Fruit Tree is a framework that abstracts tree structures across combinatorics, algebra, geometry, and robotics into a unified formalism.
• It enables exact modeling in mathematics through permutrees, metallic trees, and tree arithmetic, providing bijections, lattice congruences, and polyhedral realizations.
• In agricultural applications, it bridges RGB-D sensing, LiDAR, procedural rendering, and robotic control to improve fruit counting, pruning analysis, and harvesting.

“Generalized Fruit Tree” is not a single standardized object in the arXiv literature. The phrase spans several technical lineages: exact generalized tree models in combinatorics and algebra, and species-agnostic or task-agnostic representations of fruit trees in computational agriculture. In the first lineage, permutrees unify permutations, binary trees, Cambrian trees, and binary sequences; metallic trees generalize Fibonacci trees; and several arithmetic formalisms treat rooted trees as canonical carriers of number systems (Pilaud et al., 2016, Margenstern, 2019, Tarau, 2013, Luccio, 2015). In the second, fruit trees are represented as RGB-colored 3D point clouds, procedural synthetic scenes, LiDAR-derived pruning structures, neural radiance fields for multi-fruit counting, and perception-and-control targets for robotic harvesting (Fu et al., 2023, Duboudin et al., 2019, Westling et al., 2021, Meyer et al., 26 May 2025, Gursoy et al., 2023, Oh, 2024). This suggests that the most rigorous interpretation of the term is comparative: a generalized fruit tree is a tree representation that abstracts away from a single topology, species, or task while preserving the structure needed for combinatorial, algebraic, geometric, or robotic inference.

1. Permutrees as a unified generalized tree model

In combinatorics, the most explicit formal answer to a generalized tree structure is the permutree. A permutree is a directed tree $\tree$ with vertex set $V$ endowed with a bijective vertex labeling $p:V\to [n]$, such that each vertex has one or two parents and one or two children, and any binary branching or merging must respect a left/right separation of smaller and larger labels relative to $p(v)$ (Pilaud et al., 2016). The model is parameterized by decorations recording whether a vertex behaves locally like a path node, a downward binary branching node, an upward binary merging node, or both.

The importance of the formalism lies in exact specialization. For $\delta=\texttt{none}^n$, the structure is a directed path and $PT(\texttt{none}^n)$ corresponds to $S_n$. For $\delta=\texttt{down}^n$, one obtains rooted planar binary trees with inorder labeling. For $\delta\in\{\texttt{down},\texttt{up}\}^n$, the objects are precisely $\delta$-Cambrian trees. For $V$0, the class corresponds to binary words of length $V$1 (Pilaud et al., 2016). The model therefore does not merely resemble these classical families; it literally contains them under one local rule system.

The combinatorial bridge from decorated permutations to trees is supplied by leveled permutrees and the insertion map $V$2, which is a bijection from $V$3-decorated permutations to leveled $V$4-permutrees. The underlying map $V$5 identifies precisely the linear extensions of a given permutree. The induced permutree congruence is a lattice congruence of weak order, and increasing edge rotations generate the $V$6-permutree lattice. The resulting quotient and rotation picture unifies the weak order, the Tamari lattice, Cambrian lattices, and the boolean lattice (Pilaud et al., 2016).

The same unification persists geometrically and algebraically. The cones $V$7 form the complete simplicial $V$8-permutree fan, and the corresponding permutreehedron $V$9 specializes to the permutahedron, Loday’s associahedron, Hohlweg–Lange associahedra, certain graphical zonotopes, and a parallelepiped generated by $p:V\to [n]$0 in the all-$p:V\to [n]$1 case. On the algebraic side, the span of the basis elements $p:V\to [n]$2 is a Hopf subalgebra containing the known Hopf-algebraic structures on permutations, binary trees, Cambrian trees, and binary sequences (Pilaud et al., 2016). In this setting, a generalized fruit tree is exact, finite, decorated, lattice-theoretic, polyhedral, and Hopf-algebraic.

2. Metallic trees and generalized Fibonacci structures

A second exact use of generalized tree language appears in the theory of metallic trees, presented as generalized Fibonacci trees. These are infinite finitely generated rooted trees with black and white nodes governed by the rules

$p:V\to [n]$3

with $p:V\to [n]$4 (Margenstern, 2019). A black node has exactly $p:V\to [n]$5 children, one black and $p:V\to [n]$6 white; a white node has exactly $p:V\to [n]$7 children, one black and $p:V\to [n]$8 white. The theory distinguishes the white metallic tree $p:V\to [n]$9, rooted at a white node, from the black metallic tree $p(v)$0, rooted at a black node.

The model generalizes the classical Fibonacci-tree case $p(v)$1, where the rules become $p(v)$2 and $p(v)$3. It is tied to the hyperbolic tilings $p(v)$4 and $p(v)$5: the white metallic tree spans a sector of each tiling, while the black metallic tree spans a strip. The level sizes satisfy the second-order recurrences

$p(v)$6

$p(v)$7

with characteristic polynomial $p(v)$8 (Margenstern, 2019).

A major contribution is the numeration system built from the metallic sequence. Every positive integer can be written as

$p(v)$9

and canonical metallic codes are obtained by forbidding the pattern $\delta=\texttt{none}^n$0. These codes support arithmetic, normalization, increment, decrement, comparison, and tree navigation. Particularly notable coordinate identities are

$\delta=\texttt{none}^n$1

The paper also proves linear-time path algorithms for sectors and strips (Margenstern, 2019).

The white-rooted and black-rooted cases differ sharply. In $\delta=\texttt{none}^n$2, every node has a preferred son with code $\delta=\texttt{none}^n$3, and the preferred son is locally identifiable. In $\delta=\texttt{none}^n$4, that direct property fails; it is replaced by a successor structure in which the node with code $\delta=\texttt{none}^n$5 is not necessarily a child. For a node $\delta=\texttt{none}^n$6 that is not rightmost on its level, the successor is the leftmost child of $\delta=\texttt{none}^n$7; for the rightmost node of a level, the successor is the rightmost node on the next level (Margenstern, 2019). A generalized fruit tree in this line is therefore an infinite, typed, recursively generated tree coupled to hyperbolic geometry and a nonstandard positional numeration.

3. Trees as arithmetic carriers

A different tradition treats generalized trees not as models of order or tilings, but as the primary objects of arithmetic. One formulation uses the free algebra of ordered rooted binary trees with empty leaves, $\delta$3 together with the signatures $\delta=\texttt{none}^n$8, $\delta=\texttt{none}^n$9, and $PT(\texttt{none}^n)$0 for three free algebras isomorphic to $PT(\texttt{none}^n)$1 (Tarau, 2013). The map from trees to naturals is

$PT(\texttt{none}^n)$2

with inverse obtained from the factorization $PT(\texttt{none}^n)$3. Arithmetic is then defined directly on tree terms through generalized constructors and deconstructors such as $PT(\texttt{none}^n)$4, $PT(\texttt{none}^n)$5, $PT(\texttt{none}^n)$6, and $PT(\texttt{none}^n)$7, allowing comparison, addition, subtraction, multiplication, exponentiation, division with remainder, gcd, and lcm to be written by structural recursion and pattern matching (Tarau, 2013).

The same paper extends the representation to signed rationals via the Calkin–Wilf bijection. Positive rationals are encoded as pairs $PT(\texttt{none}^n)$8, with

$PT(\texttt{none}^n)$9

and the inverse recursion

$S_n$0

Here the generalized tree is a canonical algebraic representation of $S_n$1 and $S_n$2, not a botanical object (Tarau, 2013).

A related but distinct arithmetic is defined on non-empty rooted unordered trees. In that system, $S_n$3 denotes the one-vertex tree, addition $S_n$4 merges the roots, multiplication $S_n$5 merges the root of $S_n$6 with each vertex of $S_n$7, and stretch $S_n$8 adds a new root above $S_n$9. The vertex-count relations are

$\delta=\texttt{down}^n$0

Addition is commutative and associative, multiplication is associative but generally not commutative, and

$\delta=\texttt{down}^n$1

The paper defines add-prime and mult-prime trees, proves that mult-primality is decidable in polynomial time, and establishes unique multiplicative factorization (Luccio, 2015).

Taken together, these formalisms show that a generalized fruit tree can mean a tree-valued arithmetic substrate. This suggests a broad technical pattern: trees can be generalized either by enlarging their local combinatorics, as in permutrees and metallic trees, or by reinterpreting the tree itself as the element of an algebraic universe.

4. Geometric, procedural, and pruning-oriented fruit-tree representations

In agricultural perception, generalized fruit-tree models are usually geometric rather than topological. A fusion-driven reconstruction framework uses a handheld platform with a VectorNav VN-100 IMU, a Velodyne Puck Lite LiDAR, an Intel RealSense D405 RGB camera, a FLIR Vue thermal camera that is not used in the study, and an onboard Jetson TX2 computer. LiDAR and IMU are fused with Fast-LIO2 to construct the geometric structure of the map, and RGB imagery is rendered onto the map to produce a dense, 3D RGB-colored point cloud. Fruit positions are attached through manual labeling, with future localization envisaged through a re-trained YOLOv5 network and 2D-to-3D projection. The representation is therefore a colored 3D point cloud with localized fruit positions, not a topological tree graph, branch skeleton, mesh, voxel occupancy map, organ-level semantic graph, or parametric architectural model. In orchard comparison against a RIEGL VZ-1000, registration was assessed using the average ratio metric, yielding $\delta=\texttt{down}^n$2 with thresholds $\delta=\texttt{down}^n$3 (Fu et al., 2023).

A procedural rendering framework addresses generalization from the opposite direction: not sensing real trees, but generating synthetic ones. That framework uses Blender and Python, builds a naked tree from Weber and Penn tree-generation rules through a Blender add-on, then adds fruits and leaves randomly and uniformly according to densities. The scene is centered in a spherical HDRI environment, and camera generation is controlled by sampling origin $\delta=\texttt{down}^n$4 from a hollow external cylinder $\delta=\texttt{down}^n$5 and target point $\delta=\texttt{down}^n$6 from an internal cylinder $\delta=\texttt{down}^n$7. Each time-step produces an RGB image and a ground-truth semantic segmentation map. Reported output size is $\delta=\texttt{down}^n$8, and rendering speed is around 10–30 seconds per pair using GPU rendering on NVIDIA GTX 1080 (Duboudin et al., 2019). The framework is generalized in the sense of controllable data generation and domain randomization, not in the sense of a biologically complete plant model.

A third geometric line treats the fruit tree as a LiDAR-derived pruning object. In a framework for automated pruning suggestion, the tree is voxelized, represented as a graph of occupied voxels rooted at the trunk, and evaluated by a score

$\delta=\texttt{down}^n$9

where $\delta\in\{\texttt{down},\texttt{up}\}^n$0 measures light distribution, $\delta\in\{\texttt{down},\texttt{up}\}^n$1 is normalized volume, and $\delta\in\{\texttt{down},\texttt{up}\}^n$2 is normalized total absorbed light. The score showed $\delta\in\{\texttt{down},\texttt{up}\}^n$3 against fruit count for avocado and $\delta\in\{\texttt{down},\texttt{up}\}^n$4 for mango. Pruning simulation, validated experimentally, achieved an average F1 score of 0.78 across 144 experiments, and light distribution was improved by up to 25.15%, demonstrating a 16% improvement over commercial pruning on a real tree (Westling et al., 2021). Here the generalized fruit tree is a structural-radiative point-cloud object whose value lies in comparative pruning analysis.

5. Neural fields, counting, and robotic harvesting

Generalization in orchard computer vision increasingly appears as cross-fruit, multi-view, and action-oriented representation learning. FruitNeRF++ counts fruits from unstructured orchard photographs by combining contrastive learning with a neural radiance field augmented by semantic and instance outputs. The model includes a density field $\delta\in\{\texttt{down},\texttt{up}\}^n$5, an appearance field $\delta\in\{\texttt{down},\texttt{up}\}^n$6, a semantic field $\delta\in\{\texttt{down},\texttt{up}\}^n$7, and an instance field $\delta\in\{\texttt{down},\texttt{up}\}^n$8, with volumetric rendering used for RGB, semantics, and instance embeddings. The extracted 3D representation is a fruit point cloud of pairs $\delta\in\{\texttt{down},\texttt{up}\}^n$9, clustered using HDBSCAN with distance

$\delta$0

The method is evaluated on a synthetic dataset containing apples, plums, lemons, pears, peaches, and mangoes, and on the FUJI apple benchmark. Reported average F1 is 0.925 with ground-truth masks, 0.832 with Grounded-SAM masks, and 0.776 with Detic masks on synthetic data; on FUJI, FruitNeRF++ achieves $\delta$1, compared to $\delta$2 for Gené-Mola et al. (Meyer et al., 26 May 2025). In this setting, the generalized fruit tree is a multi-view implicit field whose fruit instances are encoded by learned 3D embeddings rather than fruit-specific shape templates.

Robotic harvesting introduces a different abstraction. A dual-arm harvesting robot based on the BAZAR mobile cobot uses two 7-DOF Kuka LWR4 arms, an RGB-D camera, a cutting tool, and a collecting tool. Perception combines YOLOv5 fruit detection, DeepSORT tracking, RGB-D 3D localization of fruits, and trunk localization via HSV segmentation plus depth thresholding. Control is cast as Hierarchical Quadratic Programming with hard constraints for robot joint limits, robot self-collisions, robot-fruit collisions, and robot-tree collisions. The system was validated on 1935 RGB-D images from an artificial orange tree; among 266 occluded images, fruit ID assignment was correct in 83%, and in the remaining non-occluded images it was 100% accurate. In one experiment, 5 oranges were detected, 1 was missed due to heavy occlusion by leaves, and all detected fruits were harvested in 32 seconds; in a second experiment, one detected orange was unreachable (Gursoy et al., 2023). The generalized fruit tree here is not a full botanical model but a perception-and-constraint environment that supports target localization, obstacle modeling, and coordinated manipulation.

These two lines differ in emphasis. FruitNeRF++ is fruit-agnostic at the counting stage because instance identity is embedded into a neural field; the dual-arm harvesting system is generalized mainly at the level of task decomposition, RGB-D perception, and optimization-based collision handling (Meyer et al., 26 May 2025, Gursoy et al., 2023). This suggests two complementary strategies for generalization: learn a fruit-agnostic 3D representation, or define a task-agnostic manipulation architecture.

6. Limits of generalization and the current research agenda

A large review of front-view fruit-tree image segmentation makes the central limitation explicit: the field still lacks “a versatile dataset and segmentation model that could be applied to a variety of tasks and environments.” The review covers 158 papers, of which 76 are rule-based and 82 are deep-learning-based, and organizes them by method, image type, task, fruit species, and year. It argues that current methods are highly specific to a given task and environment, that public datasets are sparse and narrow, and that only 11 public datasets are available in the reviewed scope (Oh, 2024).

The review also identifies the dominant axes of fragmentation: task specificity, environment specificity, species imbalance, labeling inconsistency, and restricted evaluation across orchards, seasons, and training systems. Apples and grapes are the two most studied fruits; RGB and RGB-D dominate current deep-learning practice; and phenotyping and harvesting are the most common tasks. For generalized fruit-tree perception, the recommended directions are monocular depth estimation, CNN-transformer fusion, development of a versatile unified agriculture model, deep-learning-based 3D reconstruction of tree rows, construction of versatile large high-quality datasets, and increased use of few-shot and self-supervised learning (Oh, 2024).

Placed alongside the formal exact models of combinatorics and arithmetic, this agricultural diagnosis is revealing. In permutrees, metallic trees, and tree arithmetic, generalization is achieved by exact rules, exact bijections, and exact structural theorems (Pilaud et al., 2016, Margenstern, 2019, Tarau, 2013, Luccio, 2015). In orchard robotics and vision, generalization is instead partial and empirical: a point cloud may be species-agnostic yet lack topology; a procedural renderer may be flexible yet not botanical; a neural field may be fruit-agnostic yet sensitive to pose and mask quality; and a harvesting controller may be modular yet still rely on crop-specific geometric assumptions (Fu et al., 2023, Duboudin et al., 2019, Meyer et al., 26 May 2025, Gursoy et al., 2023).

A plausible implication is that “Generalized Fruit Tree” currently names a research objective more than a settled formalism. In exact mathematics, the objective has already produced unified tree models. In agriculture, it remains an open synthesis problem: to combine geometry, semantics, topology, organ relations, sensing, and task constraints into a representation that is simultaneously species-agnostic, task-flexible, and operationally robust (Oh, 2024).