Fibonacci and Trees

written by Deborah S.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 

You may recognize the numbers above as the famous Fibonacci Sequence.

Developed by Leonardo Bonacci –  that’s right, his name is not Fibonacci; Fibonacci is short for filius Bonacci, which means “son of Bonacci,” and was applied by historians to distinguish this Leonardo from another born in his hometown of Pisa – this sequence begins with a 1 and another 1, then combines those numbers to produce the next, after which the process continues. So, in the first iteration, 1 + 1 = 2; then 2 + 1 = 3; 3 + 2 =  5, and so on.

This sequence is a favorite among many, even outside of the realm of mathematicians, partially because it is relatively simple to grasp and a fun concept but also because it reportedly appears often in nature.

If a number appears somewhere in the sequence, we might refer to it as a Fibonacci number. As it turns out, Fibonacci numbers occur often in the physical world. For example, the number of petals on a flower is frequently a Fibonacci number.  Bee pedigrees – the number of parents, grandparents, great-grandparents, etc. – follow the Fibonacci sequence by generation because male bees have only one parent. Most spirals seen in nature can be approximated by using a Fibonacci spiral, pictured in the Italian stamp below.

Fibonacci, his sequence, and the spiral on an Italian postage stamp

Today, however, let’s focus on trees! Specifically, how the branching of trees seems to be related to the Fibonacci sequence!

Let’s say one wanted to draw a tree. The sequence is a wonderful tool for crafting a realistic-looking tree. Begin with one trunk, which splits into two. Then, proceed with only one of those two branches splitting, so now you have three branches at the tree’s height.  Next, have two of those branches split, giving you five at the next level. Then. three of those five lanes split, making eight branches. And so on. At any given height, you have a Fibonacci number of branches.  Also, notice that how many branches you have split at a given height is equal to the previous Fibonacci value: meaning when we had five branches and wanted to divide again, we diverged three of those branches, since the previous Fib number was three. This method produces a rather convincing tree:

Fibonacci tree - graphic from Botanica Mathematica

You can even utilize the Fibonacci numbers in the width of the trunk and branches. An enterprising creative used this notion to knit a Fibonacci tree. Using 21 – the eighth Fibonacci number – as the trunk’s thickness, the next two branches became widths of eight and 13, the numbers that combined to create 21.  Then, those branches were reduced into the preceding Fib values until the upper reaches were left with a width of one.

A knitted Fibonacci tree - image from Botanica Mathematica

The sequence is not just the domain of fictional trees.  

The numbers appear in real trees, albeit in a different way.  How they are presented in real-world trees is most easily described with a specific example. Imagine standing at the base of an oak and looking up. You will see branches sticking out from the trunk at various heights.  The number of branches does not follow the sequence but they spiral in a specific pattern. Botanists call this pattern a tree’s phyllotaxis.  In 1754, Charles Bonnet discovered the pattern could be explained as a fraction or ratio. As it turns out, the ratios are composed of numbers in the Fibonacci sequence. For our oak tree, the phyllotaxis can be described as a fraction of 2/5, meaning five branches will spiral around the trunk in two rotations.  If you were to use the branches like stairs, starting at branch one, and going up to two, then to three, four, and finally five, on the fifth branch you would have walked around the trunk of the tree 2 times. Then, the pattern repeats, all the way to the top!  

Other tree types follow similar routines; elms have a 1/2 pattern, beeches 1/3, willows 3/8, and almond trees 5/13. Notice that all the numbers in these ratios are of the Fibonacci variety: 1, 2, 3, 5, 8, and 13. The leaves also follow the same pattern, spiraling around the branches, as in the image below.

Fibonacci leaf spiraling - photo from Brown University Mathematics

That these numbers pop up in this phenomenon is intriguing, but the why behind it is even more interesting: sunlight!

In 2011, an ingenious youngster hammered home the question: what were you doing in seventh grade? His answer was winning a Young Naturalists’ Award for figuring out why branches and leaves follow Fibonacci patterns. He devised an experiment that illustrates how this patterning maximizes the absorption of sunlight. He built a mini tree using the oak’s 2/5 ratio and attached solar panels instead of leaves. He then positioned this makeshift tree and a human-made solar array (i.e. flat, angled panels) outside and recorded the energy absorbed through the two set-ups. The Fib tree absorbed about 25% more energy! This reading leads to the conclusion that trees might spiral how they do to maximize photosynthetic capabilities.

The tree array's energy absorption - graphic by Aiden Dwyer
Energy absorbtion by a standard solar array - graphic by Aiden Dwyer

Being a math and nature lover, I am fascinated by cases where they intertwine. Do our mathematical tendencies come from what we see in the world, or do we apply our mathematical ideas to what we observe? Either way, these overlaps fill me with a desire to learn!

Further Reading and Exploration

Fibonacci Tree – Botanica Mathematica

The Secret of the Fibonacci Sequence in Trees – American Museum of Natural History

Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur on Plants by Irving Adler

Spirals and phyllotaxis – Princeton University

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