Recursive Tree Generator (3D)

What the 3D scene shows

The same generator as the 2D page, but each child is turned about an axis that rotates around its parent by the azimuth step (137.5° gives the golden-angle spiral seen in many plants), so the crown fills space. Every branch is a Blender-modelled bark segment scaled to its length and diameter; the leaves are Blender leaf clusters placed at the tips. Sun elevation moves the shadows; wind bends every level in proportion to its depth.

Recursive branching

A tree of depth n starts from a trunk of length 1. Every branch of length L carries b children of length ri L, turned by angles θi. The number of branches is (bn+1 − 1)/(b − 1) and, with a common ratio r, the total wood length is the geometric series ∑ (b r)k. If b r > 1 the length grows without bound as the depth increases.

Leonardo da Vinci's rule and Murray's law

Leonardo noted that the cross-section of a trunk equals the sum of the cross-sections of its branches, i.e. dparent2 = ∑ dchild2; measurements on real trees agree roughly with exponents close to 2. For blood vessels Murray's law gives exponent 3 (minimum pumping plus metabolic cost for laminar flow). The page lets the exponent k vary and measures the residual on every parent of the generated tree.

Fractal dimension

A symmetric tree is self-similar: b copies scaled by r, so the similarity dimension is D = ln b / ln(1/r). The box-counting estimate counts occupied boxes N(ε) of the tip set and fits ln N against ln(1/ε). For large r or wide angles the crowns overlap in the plane and the measured value saturates below the similarity dimension (a 2D projection cannot exceed 2); the check therefore uses a canonical sparse tree (r = 0.6).

Simplifications

Real trees respond to light, wind and damage; here randomness is a uniform jitter of angles and lengths, tropism a constant upward bias of each direction, and wind a sinusoidal angle offset growing with the level. Leaves are decorative markers and carry no photosynthesis model.