A couple of years ago I was doodling a bunch of little flowers. My intention was not to do math, far from it. However, I realized that by making up arbitrary rules, like fixing the growth rate of each flower, these little tree-like structures were full of math. In particular, they demonstrate exponential growth and model how the three math operators: exponents, logarithms and radicals are closely related together.
Most good things start out as a doodle on a piece of paper. For example, the drawing of one of these trees is shown below. The tree starts at an origin and at each step, the tree branches multiply by a factor of .
The process for drawing this tree is described below:
- Start by drawing a point that forms the base of the flower tree. This point is considered level zero or in other words we would say that this tree has a height of zero.
- Draw branches starting from the point outward. The number of branches sprouting from the base represents the growth rate of the tree. In this case, the growth rate is equal to .
- Repeat this process for each tip of the branchs drawn as if they were the base of the tree. Each new level follows the same process until the tree (or flower!) is the desired height.
- Finally, draw some decorative leaves at the top level of the tree.
More formally, the tree structure has three important properties: the number of leaves on the tree, the height of the tree and growth rate of the tree. These properties are highlighted in figure above and summarized in the table below.
The number of nodes on the top-most level of the tree. The leaves can also be though of as the total population.
The number of levels the tree where the starting point of the tree is considered level zero. The levels can also be thought of as time elapsed.
The growth rate, sometimes referred to as the base, describes the branching factor at each level of the tree.
The interactive graphic below allows you to generate many different exponential trees. The first slider controls the growth rate of the tree and the second slider controls the number of levels of the tree.
By increasing the levels of the tree, the number of leaves grows in an exponential manner. At each new level of the tree the number of leaves on the next level is equal to the current population of leaves multiplied by the growth rate factor. In contrast, linear growth would add a static amount at each step.
Each of the three operators: the exponent, logarithm and radical operator, correspond to a property of the tree. Specifically, given two other properties as input each operator returns the missing third property of the tree. Each relationship the operators have to the tree is summarized in the table below.
Given the branching factor and the number of levels of the tree, the exponent operator returns the number of leaves on the tree.
Given the values for the branching factor and leaves of the tree, the logarithm operator returns the levels of the tree.
Given the levels of the tree and the number of leaves, the radical operator returns the growth rate of the tree.
Shown below are some examples of trees with unique properties that demonstrate these operator relationships.
In conclusion, flowers, trees and other plants demonstrate exponential growth. Of course, the natural world is a lot more messy, but hopefully this technique gives a basic understanding of how exponential growth can be modeled using math.
I hope you have enjoyed this article. If you have any feedback, draw any of your own exponential trees or decide to use this as a reference for students I would love to know! Send me an email at firstname.lastname@example.org.