Unifying constructal theory of tree roots, canopies and forests

Here, we show that the most basic features of tree and forest architecture can be put on a unifying theoretical basis, which is provided by the constructal law. Key is the integrative approach to understanding the emergence of "designedness" in nature. Trees and forests are viewed as integral components (along with dendritic river basins, aerodynamic raindrops, and atmospheric and oceanic circulation) of the much greater global architecture that facilitates the cyclical flow of water in nature (Fig. 1) and the flow of stresses between wind and ground. Theoretical features derived in this paper are: the tapered shape of the root and longitudinally uniform diameter and density of internal flow tubes, the near-conical shape of tree trunks and branches, the proportionality between tree length and wood mass raised to 1/3, the proportionality between total water mass flow rate and tree length, the proportionality between the tree flow conductance and the tree length scale raised to a power between 1 and 2, the existence of forest floor plans that maximize ground-air flow access, the proportionality between the length scale of the tree and its rank raised to a power between -1 and -1/2, and the inverse proportionality between the tree size and number of trees of the same size. This paper further shows that there exists an optimal ratio of leaf volume divided by total tree volume, trees of the same size must have a larger wood volume fraction in windy climates, and larger trees must pack more wood per unit of tree volume than smaller trees. Comparisons with empirical correlations and formulas based on ad hoc models are provided. This theory predicts classical notions such as Leonardo's rule, Huber's rule, Zipf's distribution, and the Fibonacci sequence. The difference between modeling (description) and theory (prediction) is brought into evidence.