樹冠的形狀大部份主要是由陽光和重力造成的，有些樹樹冠取決於這兩者的平衡而有特定形狀Tree crowns grow into self-similar shapes controlled by gravity and light sensing
αp表示光照參數，αg表示重力參數，曲線下方深綠色的自然界理論上不會出現，因為這些樹冠形狀光照與重力之間狀態不會達成平衡
A set of crown shapes obtained from a numerical computation, varying αg and αp, and with γ = 0.01. Green shapes (above the curve) have converged towards a self-similar-one, whereas grey shapes have not converged. For each shape, the parameter values correspond to the grid intersection. (Online version in colour.)
Sketch of a growing tree crown. h and v are unit horizontal and vertical vectors, respectively, n and t are the unit vectors, respectively, normal and tangent to the front and the unit vector ℓ points towards the mean direction of light, which is the first bisector of 
. The angle ψ = φ + π/2 represents the local amount of sunlight intercepted for this axisymmetric shape. The inset shows a zoom around the front to highlight the conditions for self-similarity of the growing shape. (Online version in colour.)
. The angle ψ = φ + π/2 represents the local amount of sunlight intercepted for this axisymmetric shape. The inset shows a zoom around the front to highlight the conditions for self-similarity of the growing shape. (Online version in colour.)
Figure 2. Superimposed views of a tree crown growth for two different initial conditions. In (a), the initial shape is a circle and in (b) a circle with a significant 
perturbation. Parameters are: αg = αp = 0 and γ = 0.01. The light green curves correspond to different instants. The dark-green curves correspond to the self-similar solution after a long period. The shape is rescaled at each time step in order to keep its volume constant equal to one. (Online version in colour.)
perturbation. Parameters are: αg = αp = 0 and γ = 0.01. The light green curves correspond to different instants. The dark-green curves correspond to the self-similar solution after a long period. The shape is rescaled at each time step in order to keep its volume constant equal to one. (Online version in colour.)
Attracting shapes obtained for different values of γ: γ = 0.04, 0.02, 0.01, 0.005 and 0.0025, with αg = αp = 0 (a) and αg = 0.8, αp = − 1 (b). The envelope of the grey lines given by equation (4.4) is the expected self-similar shape. When γ → 0, the time-evolving shapes converge towards this self-similar shape. (Online version in colour.)
Figure 5. (a) Trajectories of points along the front, for the cases αg = αp = 0 and (b) αg = αp = 2. Starting from the self-similar profile corresponding to t = 1, 31 points are advected backward in time using equation (2.1) until t = 0.1.
Figure 7. Comparison between real tree crowns and self-similar shapes of the model: (a) Betula pubescens, αg = 0.927, αp = 0.267, d = 0.022; (b) Quercus castaneifolia, αg =−1.00, αp = 0.353, d = 0.019; (c) Enterolobium cyclocarpum, αg =−0.490, αp =−0.488, d = 0.023 (black curve: αg =−1.00, αp = 0.034, d = 0.020), (d) Thuja occidentalis, αg = 0.379, αp =−1.23, d = 0.018 (black curve: αg = 5.00, αp = 2.69, d = 0.079). The red curve represents the best fit and the black curve the best fit with αp > 0. See the electronic supplementary material for comparisons with other tree species. (Online version in colour.)
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