Statistical hydraulic model for the Leonardo's rule

More than five hundred years ago Leonardo Da Vinci found a pattern in the growth of trees nowadays known as the Leonardo's rule. This rule relates the thickness of the stem with the thickness of the branches at different bifurcation stages in a Pythagorean fashion. He argued that his rule was the result of the conservation of sap flux. In the present work, we explore this idea by assuming that the sap flux through each xylem element behaves as a non-ideal fluid and the size-distribution of the xylem elements obeys a power law distribution. We find that the simultaneous fulfillment of Leonardo's rule and the conservation of the sap flux, lead to a global behavior of the sap like that of an ideal fluid after summing over all xylem elements. These results are supported by field and experimental work. In particular, we corroborated the Leonardo's rule in different tree species by measuring the stem-branches thickness at different bifurcations stages. We also determined the statistical size-distribution of the xylem elements through a maceration process, finding that it corresponds to a power law distribution with an exponent close to three, which is the exponent required for the Leonardo's rule. As far as we know this is the first time that a statistical hydraulic model supported by experimental data is presented for the Leonardo's rule.