Laudator Temporis Acti

Joseph Addison, The Spectator No. 414, Wednesday, June 25, 1712:

I do not know whether I am singular in my Opinion, but, for my own part, I would rather look upon a Tree in all its Luxuriancy and Diffusion of Boughs and Branches, than when it is thus cut and trimmed into a Mathematical Figure.

Trees, in the diffusion of their boughs and branches, obey their own mathematical laws, which various investigators have attempted to discover. One of the chapters in Bernd Heinrich's The Trees in My Forest (New York: Cliff Street Books, 1997) has the title "Tree Geometry and Apical Dominance." Leonardo da Vinci also attempted to discover a mathematical rule that describes the growth of trees. See his Notebooks, Vol. I, Part VIII (Botany for Painters and Elements of Landscape Painting, tr. Jean Paul Richter):

394.
All the branches of a tree at every stage of its height when put together are equal in thickness to the trunk [below them].

All the branches of a water [course] at every stage of its course, if they are of equal rapidity, are equal to the body of the main stream.

395.
Every year when the boughs of a plant [or tree] have made an end of maturing their growth, they will have made, when put together, a thickness equal to that of the main stem; and at every stage of its ramification you will find the thickness of the said main stem; as: i k, g h, e f, c d, a b, will always be equal to each other; unless the tree is pollard—if so the rule does not hold good.

Melvin T. Tyree and M.H. Zimmermann, Xylem Structure and the Ascent of Sap, 2nd ed. (Springer, 2002), p. 143, wrote:

Although there is no evidence that Leonardo ever did any measurements to confirm this remarkable observation, botanists are well aware of the approximate correctness of this statement; measurements were made around 1900 to investigate the significance of stem dimensions in satisfying both mechanical and hydraulic demands....

Tyree and Zimmermann cite the following earlier studies:

• K. Metzger, "Studien über den Aufbau der Waldbäume und Bestände nach statischen Gesetzen," Mündener Forstliche Hefte 5 (1894) 61-74, 6 (1894) 94-119, 7 (1895) 45-97
  
• Paul Jaccard, "Eine neue Auffassung über die Ursachen des Dickenwachstums," Naturwissenschaftliche Zeitschrift für Forst- und Landwirtschaft 11 (1913) 241—279
  
• Paul Jaccard, Nouvelles recherches sur l'accroissement en épaisseur des arbres. Essai d'une théorie physiologique de leur croissance concentrique et excentrique (Lausanne-Genève: Payot, 1919)
  
• Eduard Rübel, "Experimentelle Untersuchungen über die Beziehungen zwischen Wasserleitungsbahn und Transpirationsverhältnissen bei Helianthus annuus L.," Beihefte zum Botanischen Centralblatt 37 (1920) 1-62

More recently Henry S. Horn and his students have been studying the extent to which Leonardi da Vinci's rule accurately describes the growth of trees. See a paper by one of those students, Rizwan Aratsu, "Leonardo Was Wise: Trees Conserve Cross-Sectional Area Despite Vessel Structure," Journal of Young Investigators 1 (Dec. 1998).

Related post: Tree Geometry: Parallel Lines.