# How does the number of leafs on a tree grow with the height of the tree?

As a was lying on my back in the University park to day, resting a bit between reading papers (more on that later), I looked up at a pine tree and asked myself a very computer sciencey question. How does the number of needles L on the tree grow, as the tree grows taller? If the height of the tree is h, what is the size of L?

## Exponential?

I initially thought that surely this must be exponential, after all just think of a binary tree: The number of leaf nodes in a balanced binary tree of height h is $2^h$. So maybe:

$L \approx 2^h$

## Polynomial?

Then I thought some more. A tree looks somewhat like a sphere, and the density of leafs within the crown of the tree seems somewhat independent of the tree height. So maybe the number of leaves (or pine needles) grows like the volume of a sphere, i.e. 4/3*pi*(h/2)^3, which is a third degree polynomial function. If the leaves are primarily on the surface of the crown, with inner leaves being shaded away, the number of leaves is more like the surface of a sphere, i.e. 4*pi*(h/2)^2, a second degree polynomial function. Maybe the answer is in between, with most of the leaves on the surface where the sun is, and a decreasing number of leaves as one approaches the trunk of the tree. So maybe:

$4 \pi \frac{h}{2}^2 \leq L \leq \frac{4}{3} \pi \frac{h}{2}^3$

## Simply looking it up...

Thinking about stuff is all well and good, but sometimes looking in the back of the book is better. The following webpages try to answer the question, by looking at the area beneath a tree that is covered by leaves in the fall (some estimate this to be 4 times the area directly beneath the tree) and then multiplying the area by the number of leaves that are observed within a square meter of the area beneath the tree. This obviously doesn't account for leaves that have been blown away by the wind, and that the density of the leaf-cover might vary with proximity to the trunk.

• http://wetlandnotes.com/estimating-the-number-of-leaves-on-a-tree
• http://www.walterreeves.com/gardening-q-and-a/tree-leaves-calculating-the-number/

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