Leonardo da Vinci had a rule for depicting trees in winter, which recognises the nutrient-transporting function of tree branches.

Da Vinci described the rule in his Manuscript M, foglio 78v (1490-1500): At the bifurcation of the trunk into the first two branches, the combined cross-sectional area of the branches equals the cross-sectional area of the trunk. The same rule applies at each further bifurcation of a branch into smaller branches, all the way to the thinnest twigs. This fractal pattern preserves the total cross-sectional area of the tree regardless of whether it cuts across the trunk, the main branches or the crown.

Expressed as a formula for two branches with radii a and b bifurcating from a trunk of radius c, the rule stipulates: a² + b² = c². A variant of da Vinci’s rule preserves radius, and equally circumference, instead of area: a + b = c. An alternative scaling instead preserves volume: a³ + b³ = c³. Generalising for any scaling exponent α gives: $a^{\alpha }$ + $b^{\alpha }$ = $c^{\alpha }$.

What value does α take in nature? Trees in nature generally have α values of 2 to 3, according to Gao and Newberry (2025). Their grove below shows randomly generated representations of trees, with α = 1 (left), α = 2 (middle) and α = 3 (right). Refresh the page to generate a new grove.