How the shape of the tree crown is related to tree growth is still by and large unknown. Intuitively the crown shape should influence the loss of biomass during growth, as top-heavy crown shapes require the loss of branches at lower heights.
Here we theoretically investigate how the loss of biomass during growth influences crown-rise efficiency based on the assumption that tree growth is isometric and that the leaf area is homogenously distributed with the crown. This implies that top-heavy tree crowns loose more biomass than bottom-heavy crowns. Based on this idea, we derive an analytic expression for tree crown-rise efficiency. We then investigate the trade-off between light-use efficiency and crown-rise efficiency for a range of possible crown shapes. To understand the effect of the light environment for this trade-off, we vary the sun angle and canopy vertical light gradient, and find for each such combination a range of trees that optimally balance both light-use and crown-rise efficiency, i.e., are Pareto efficient.
Our main findings are: (1) For a stand-alone tree without external shading only a low bush-like tree is Pareto efficient. (2)We only find narrow elongated shapes when light gradient is weak and sun angle is low, but conical and spherical shapes are never found, indicating that these shapes are not light-use efficient. (3) Large variations in the Pareto efficient shapes are found in intermediate light gradients, suggesting a new mechanism for coexistence. (4) In steep light gradient we find mostly top-heavy crowns. (5)Surprisingly, hourglass shapes are found in many light conditions.
Our results are largely consistent with field data from 36 datasets between Lat. 0 and Lat. 60, indicating that the tree crown length-to-height ratio is positively correlated to latitude, corroborating the common belief that top-heavy crowns are primarily found at high sun angles corresponding to low latitudes.