Wind loads and competition for light sculpt trees into self-similar structures

Abstract

Trees are self-similar structures: their branch lengths and diameters vary allometrically within the tree architecture, with longer and thicker branches near the ground. These tree allometries are often attributed to optimisation of hydraulic sap transport and safety against elastic buckling. Here, we show that these allometries also emerge from a model that includes competition for light, wind biomechanics and no hydraulics. We have developed MECHATREE, a numerical model of trees growing and evolving on a virtual island. With this model, we identify the fittest growth strategy when trees compete for light and allocate their photosynthates to grow seeds, create new branches or reinforce existing ones in response to wind-induced loads. Strikingly, we find that selected trees species are self-similar and follow allometric scalings similar to those observed on dicots and conifers. This result suggests that resistance to wind and competition for light play an essential role in determining tree allometries.

Fig. 1

Principles of MECHATREE. a Each virtual tree is an assembly of different units: segments, foliages, seeds and a reserve. b Primary growth relies on the reserve to grow new segments and seeds. Secondary growth costs include maintenance in addition to diameter growth of each branch segment. c Formal neural networks are used to model the biochemical regulation of growth. d The growth of new segments follows a rule based on three angles: θ ₁, θ ₂ and γ. e Illustration of Strahler ordering of branches (Methods section)

Fig. 2

Evolution of a forest on a virtual island. a Example of the evolution of a virtual forest over 10,000 yrs (Supplementary Movie 1). Initially, 20,000 trees with random genomes are sown. Each circle is a tree whose centre is at the centre of gravity of foliages, and whose radius is the standard deviation of the foliage distribution projected onto the ground. The outer black circle of radius R = 200L is the habitat limit. Colours are determined using three neutral marker genes as RGB values. Because trees growing at the periphery are on average larger than the others, allometric statistics (see below) are performed on trees with their trunk in a central zone of radius 0.9R. b Number of trees and number of species as a function of time, for the same forest as in a

Fig. 3

Allometric scalings. a For the 32 forests of the first round, the effective biomass $\overline{M}$ (unit L ³) is plotted as a function of the effective number of trees $\overline{N}$, for the first 500 yrs (the light blue line shows the history of a particular forest). Only 2% of the dataset is shown, but the red line shows a linear regression fit on the entire dataset with $\overline{N}$ as weight. Large excursions to the left of the regression line correspond to strong wind events during which a large number of trees can die (Fig. 2b and Supplementary Movie 1). b Each tree in the 32 forests of the first round is extracted after 3000 yrs. Their height H (unit L), crown radius C (unit L), number of foliages N and stem biomass B (unit L ³) are plotted as a function of their trunk diameter d _trunk. For clarity, only 5% of the trees are shown, but the solid lines show reduced major axis regression (RMA) on the whole dataset, with N as weight. The dashed lines show the results of the AMT model (see below)

Fig. 4

Partial view of the forest in the Final round on a small island (R = 40L), when only the fittest species remains. Only the largest tree has been coloured for clarity. In this representation, the diameter of foliage spheres is proportional to the light intercepted (Supplementary Movie 2)

Fig. 5

Self-similarity. a Representation of the Strahler order of each branch for the tree illustrated in Fig. 4. b The number of branches, their mean length (unit L), area (unit L ²) and diameter (unit L) are plotted as a function of their Strahler order for the 999-year-old tree illustrated in Fig. 4 and in a. Error bars show standard deviations around these means. Solid lines are regression fits on the first 7 ranks. Open symbols connected by dotted lines represent the same quantities when the tree is only 25 years old. c Evolution of the branching ratio R _n, length ratio R _l and fractal dimension D = lnR _n/lnR _l during the lifetime of the same tree. Grey bars show the 80% confidence interval for D. d Branch tapering is illustrated by plotting, for each segment, the average distance from the foliages, $⟨\ell ⟩$, as a function of the diameter d. The Strahler order of each segment is represented with the same colour code as in a. e Assessment of Leonardo’s rule of area conservation across branching nodes. For each node, the area ratio (i.e. the ratio between the total cross-sectional area of children segments and the parent area) is plotted as a function of the average distance $⟨\ell ⟩$ of the parent segment from the foliages. The average area ratio measured for $⟨\ell ⟩>1.5$ is 0.94. When restricted to $⟨\ell ⟩>10$, the average is 0.985

Fig. 6

Comparison of the AMT model with empirical data. a Scaling of plant height as a function of stem diameter. The allometric data of ref. for woody species are compared to least square (LS) regression $H=20.7d_{\mathrm{trunk}}^{0.538}$ , reduced major axis (RMA) regression $H=21.4d_{\mathrm{trunk}}^{0.73}$ , WBE model $H=25d_{\mathrm{trunk}}^{0.67}$ and present AMT model $H=26.7d_{\mathrm{trunk}}^{0.857}$ (Methods section and Supplementary Discussion). b Allometric data on leaf dry mass vs. trunk diameter are compared to RMA regression $M_{\mathrm{L}}=166d_{\mathrm{trunk}}^{2.17}$ , WBE model $M_{\mathrm{L}}=247d_{\mathrm{trunk}}^2$ and AMT model $M_{\mathrm{L}}=202d_{\mathrm{trunk}}^{2.14}$. c Allometric relations between leaf dry mass and stem dry mass from more than 11,000 records are compared to RMA regression $M_{\mathrm{L}}=0.113{M_{\mathrm{S}}}^{0.74}$ , WBE model $M_{\mathrm{L}}=0.176{M_{\mathrm{S}}}^{0.75}$ and AMT model $M_{\mathrm{L}}=0.124{M_{\mathrm{S}}}^{0.75}$