The area-preserving rule for botanical trees by Leonardo da Vinci is discussed in terms of a very specific fractal structure, a logarithmic fractal. We use a method of the numerical Fourier analysis to distinguish the logarithmic fractal properties of the two-dimensional objects and apply it to study the branching system of real trees through its projection on the two-dimensional space, i.e., using their photographs. For different species of trees (birch and oak) we observe the $Q^{-2}$ decay of the spectral intensity characterizing the branching structure that is associated with the logarithmic fractal structure in two-dimensional space. The experiments dealing with the side view of the tree should complement the area preserving Leonardo's rule with one applying to the product of diameter $d$ and length $l$ of the $k$ branches: $d_i{l}_i=k{d}_{i+1}{l}_{i+1}$. If both rules are valid, then the branch's length of the next generation is $\sqrt{k}$ times shorter than previous one: $l_i=\sqrt{k}{l}_{i+1}$. Moreover, the volume (mass) of all branches of the next generation is a factor of $d_i/d_{i+1}$ smaller than previous one. We conclude that a tree as a three-dimensional object is not a logarithmic fractal, although its projection onto a two-dimensional plane is. Consequently, the life of a tree flows according to the laws of conservation of area in two-dimensional space, as if the tree were a two-dimensional object.

• Received 10 October 2021
• Revised 24 February 2022
• Accepted 13 April 2022

DOI:https://doi.org/10.1103/PhysRevE.105.044412