Fractal Tree Generation

Planting the tree...

Presets

Branching rule

Wind, light and season

Leaves Hill and rocks Auto-orbit camera

Live readouts

Graphs

Branches per level (blue) and total branch length per level (orange), log scale

Cumulative branch length vs depth: measured (orange) vs closed form (dots)

Branch radius vs depth: measured (orange) vs R0 k^depth (dots), da Vinci rule

Frame time (ms), green line = 16.7 ms

Numerical checks

running...

How to use

Each branch splits into two children rotated by the branch angle and shortened by the length ratio r. The plane twist rotates the branching plane at every level (0 gives the flat 2D tree, 90 degrees a bushy 3D tree, 137.5 degrees the golden-angle spiral). Radii follow Leonardo's rule: the sum of the children's cross-section raised to the exponent equals the parent's. Drag to orbit, wheel to zoom; move the wind and season sliders to watch the canopy sway, change colour and drop its leaves.

The science

With length ratio r and two children per branch, level k holds 2^k branches of length L0 r^k, so the total length of a tree of depth D is

L(D) = L0 ((2r)^(D+1) - 1) / (2r - 1), N(D) = 2^(D+1) - 1 branches, 2^D tips

The left/right asymmetry multiplies one child by (1+a) and the other by (1-a), so each level still sums to (2r)^k L0 and the closed form is unchanged. The similarity dimension of the limit set is D_s = log 2 / log(1/r); for r = 0.74 that is 2.30, above the box dimension of a plane, which is why the 3D tree fills space far more than its 2D shadow.

Radii follow the da Vinci / Murray family: a parent of radius R splits into two children of radius R k with 2 k^alpha = 1, so k = 2^(-1/alpha). alpha = 2 (Leonardo) conserves total cross-section area, alpha = 3 is Murray's law for flow-optimal branching. The wind adds a small horizontal displacement to each branch direction, amplitude 0.06 W sin(1.6 t + 0.5 depth) (1 + 0.25 depth), so tips sway more than the trunk; this is a kinematic toy, not a structural dynamics model. Wood volume uses frusta, V = pi L (r0^2 + r0 r1 + r1^2) / 3, and 1 unit is 1 m.

Simplifications: no self-shading growth, no gravity or phototropism, leaves are decorative (their number and size are not a leaf-area model), random jitter is a seeded pseudo-random value per branch. Bark, leaf, hill and rock textures are baked in Blender 5.2; the tree itself is generated live.

What the checks mean

Count and length regenerate a clean tree (no wind, no jitter) and compare with N(D) and L(D). Connectivity decodes the rendered instance matrices and measures the largest gap between a branch's end and its children's start. da Vinci reads the rendered radii and tests 2 r_child^alpha = r_parent^alpha. Symmetry verifies that the tip centroid of a clean tree lies in the first branching plane. The ratio sweep integrates the length by explicit recursion for several r and compares with the closed form.

Keyboard and accessibility

Space pauses, R resets, C re-runs the checks (while the page has focus).
All controls are native inputs with labels.