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The mathematical treatment of leaf venation: the variation in secondary vein length along the midrib.

BACKGROUND AND AIMS
In some dicotyledonous leaves and leaflets, the secondary veins run more-or-less straight to the margins and have well-defined lengths. For a given half-lamina of length L, an equation, previously proposed, relates the lengths of these veins, p, to the distances, l, between the leaf tip and their insertions on the midrib: p = B2(x+y)l(x)(L - l)(y)/L(x+y-1), where B, x and y are fitted parameters. Aspects of the formula are re-examined, including its general applicability, significance and usefulness.


METHODS
Length measurements were made on leaves of various dicotyledons, notably Ulmus glabra, U. procera, Alnus viridis, A. glutinosa, Corylus avellana and Crataegus monogyna. Equations were fitted by non-linear regression.


KEY RESULTS
The equation has now been applied descriptively to 23 species of eight families, but it is sometimes preferable or necessary to replace the measured length, L, with a fourth parameter that may differ significantly from it. Within a given species, values of the indices x and y are positively correlated. Leaves of some U. glabra depart qualitatively from the general pattern. As an example of hypothesis testing, the equation was used to show that the retuse or emarginate leaf tips of A. glutinosa are not due to stunting. CONCLUSIONS; That the equation applies to many species suggests that the underlying processes of leaf growth are quantitatively similar. Although relevant knowledge of these is scant, consideration of mathematical relationships may help their elucidation.