Leonardo’s Rule, Self-Similarity, and Wind-Induced Stresses in Trees

Examining botanical trees, Leonardo da Vinci noted that the total cross section of branches is conserved across branching nodes. In this Letter, it is proposed that this rule is a consequence of the tree skeleton having a self-similar structure and the branch diameters being adjusted to resist wind-induced loads.

Figure 1

Two analytical models: (a) The fractal tree model. (b) The continuous tapered beam model [8].

Figure 2

Numerical tree model: (a) Sketch of the angles and unit vectors at a branching node. (b) Example of tree skeleton for ${\theta }_1=-15°$, ${\theta }_2=30°$, $\gamma =120°$, $r_1=r_2=0.75$, $p=1$, $K=10$, $D=2.41$ [as given by (9)].

Figure 3

Deterministic tree: (a) Calculated branch diameters for the skeleton shown in Fig. 2b. (b) Normalized average distance from the tips as a function of the diameter. (c) Calculated Leonardo exponent for each branching node (the horizontal bars show the mean value of $\Delta$ at each rank). In (b) and (c), the beam model corresponds to the relations (8a, 8b).

Figure 4

Stochastic tree: (a) Silhouette of a tree with parameters: ${\overline{\theta }}_1=-20°$, ${\overline{\theta }}_2=40°$, $\overline{\gamma }=100°$, ${\overline{r}}_1=0.75$, ${\overline{r}}_2=0.85$, $p=0.98$, $K=10$, $D=2.85$. (b) Corresponding Leonardo exponent [same legend as in Fig. 3c]. (c) Total cross-sectional area at every rank $k$ normalized by its maximum. The bars correspond to the tree depicted in (a), the curve is the prediction from the beam model and the symbols with the error bars correspond to the average and standard deviation on 1000 realizations where the mean branching angles are randomly taken in the intervals $-60°<{\overline{\theta }}_1<0$, $0<{\overline{\theta }}_2<60°$, $-180°<\overline{\gamma }<180°$.