3D Tree and Root Visualization¶

This work is licensed under a Creative Commons Attribution 4.0 International License.

Interactive three-dimensional visualization of branching networks, demonstrating the self-affine geometry of vascular plant architectures.

Overview¶

This application generates 3D branching structures that model:

Tree canopy architecture: Above-ground branching from trunk to twigs
Root systems: Below-ground foraging networks
Vascular networks: Internal transport systems

The visualization demonstrates how self-affine scaling creates space-filling structures that optimize resource transport.

Mathematical Model¶

WBE Branching Ratios¶

The West-Brown-Enquist model predicts specific scaling relationships for branching networks:

Radial scaling (area-preserving): [ \xi = \frac{r_{k+1}}{r_k} = n^{-½} ]

Longitudinal scaling (space-filling): [ \gamma = \frac{l_{k+1}}{l_k} = n^{-⅓} ]

where \( n \) is the branching ratio (number of daughter branches per parent).

Self-Affinity¶

Since \( \xi \neq \gamma \), the branching network is self-affine, not self-similar. Branches become relatively more slender as the network subdivides:

\[ \frac{l/r \text{ at level } k+1}{l/r \text{ at level } k} = \frac{\gamma}{\xi} = n^{1/6} \]

Features¶

Tree Generation¶

Parameter	Description	Default
Branching ratio	Children per parent	2
Length ratio	\( \gamma \)	0.7
Width ratio	\( \xi \)	0.6
Iterations	Branching levels	8
Angle spread	Branch angle variation	30°

Root Generation¶

Root systems can use different parameters reflecting: - Higher branching ratios (more fine roots) - Different length/width ratios - Gravitropism (downward bias)

Visualization Controls¶

Rotate: Click and drag to rotate the view
Zoom: Scroll wheel to zoom in/out
Pan: Right-click drag to pan
Reset: Double-click to reset view

Biological Applications¶

Canopy Architecture¶

Tree crowns exhibit characteristic scaling:

Species Type	Branching Ratio	Typical \( D_M \)
Broadleaf deciduous	2-3	1.4-1.6
Conifer	3-5	1.5-1.7
Palm	1 (unbranched)	~1.0

Root Systems¶

Root architecture varies with soil conditions:

Root Type	Description	Fractal Dimension
Taproot	Single dominant root	Lower \( D \)
Fibrous	Highly branched	Higher \( D \)
Adventitious	Surface roots	Variable

Crown Shyness¶

In dense forests, neighboring crowns maintain gaps, creating a Voronoi-like tessellation of the canopy. The 3D visualization can demonstrate this by: - Generating multiple trees - Applying collision detection - Showing the resulting canopy structure

Technical Implementation¶

Rendering¶

The visualization uses Three.js for WebGL-based 3D rendering:

// Simplified branch generation
function generateBranch(parent, level, maxLevel, params) {
    if (level > maxLevel) return;

    const length = parent.length * params.gamma;
    const radius = parent.radius * params.xi;

    for (let i = 0; i < params.branchingRatio; i++) {
        const angle = (2 * Math.PI * i) / params.branchingRatio;
        const child = createCylinder(length, radius, parent.end, angle);

        scene.add(child);
        generateBranch(child, level + 1, maxLevel, params);
    }
}

Performance Optimization¶

Level of Detail (LOD): Reduce detail for distant branches
Instanced rendering: Batch similar geometry
Culling: Skip branches outside view frustum

Analysis Tools¶

Fractal Dimension Calculation¶

The application can compute the fractal dimension of generated structures using:

Box-counting: 3D voxelization and counting
Mass dimension: Scaling of total branch volume
Surface dimension: Scaling of total surface area

Export Options¶

OBJ format: 3D mesh for external rendering
Point cloud: XYZ coordinates of branch endpoints
CSV: Tabular data of branch properties

Examples¶

Example 1: Symmetric Binary Tree¶

Parameters: - Branching ratio: 2 - Length ratio: 0.7 - Width ratio: 0.7 (self-similar) - Iterations: 10

Result: \( D \approx 2.0 \) (self-similar, fills a plane)

Example 2: Self-Affine Tree¶

Parameters: - Branching ratio: 2 - Length ratio: 0.7 (\( \gamma = n^{-1/3} \)) - Width ratio: 0.5 (\( \xi = n^{-1/2} \)) - Iterations: 10

Result: \( D_M \approx 1.5 \) (matches MST prediction)

Example 3: Dense Root System¶

Parameters: - Branching ratio: 4 - Length ratio: 0.6 - Width ratio: 0.4 - Iterations: 6 - Downward bias: 0.8

Result: Space-filling root network for nutrient foraging