# Nerding Out with Math, Nature, and Candy Lovers

October 3, 2014

Filed under A&E

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Apparently nature takes a place in all forms, including mathematics. A “tree” is a type of graph (vertices/dots connected by edges/lines) that has no loops; i.e. no path can be made from a vertex back to its original vertex. Trees, like in almost all of nature, have “leaves”. A leaf is an edge that comes off a vertex that doesn’t connect to any other vertices. Now let’s say we have multiple trees—then you guessed it—we have a forest!

In topology—which for simplicity is just a very weird mathematical world—shapes like a circle, a triangle, or even a square are considered to be the same thing because you can either pull, stretch, or squish their edges to form one another (think of it like a rubber band). Now in regards to trees, in topology the following graph would be considered to be “indistinguishable” (or the same) from the earlier example. Notice that all the vertices are still connected to each other in the same way; it’s just their orientation has shifted.

However, not every shape in topology is considered equal. For instance the following shape is distinguishable (or not the same) because unlike the previous two trees, it only has 3 vertices and 2 edges. But notice that the next tree is topologically the same.

Now for the fun part—I want you to make all the possible distinguishable trees, starting from one vertex to ten vertices to become a weird math expert!

The first three vertices I will give you because there is only one topologically distinct tree for each of them. Also remember: every edge has a vertex at each end, and you can’t make a path that starts from one vertex and can return to the same vertex, or else it isn’t a tree.

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