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Explore what a tree diagram is and how to use it to organize information and answer questions about various outcomes of a particular event in this lesson. See how to create a tree diagram through multiple examples and a detailed explanation.

Tree Diagrams

Have you ever been staring into your closet trying to figure out what to wear and wondered how many outfits you could actually put together with your different tops, bottoms, and shoes? In mathematics, we have a tool for this called a tree diagram. A tree diagram is a tool that we use in general mathematics, probability, and statistics that allows us to calculate the number of possible outcomes of an event, as well as list those possible outcomes in an organized manner.A common example used to introduce tree diagrams is to find the number of possible outcomes of flipping two coins in succession.

We know that when we flip a coin, it will either land on heads or tails, so when we flip one of the coins, we have two possible outcomes: heads or tails. When creating a tree diagram, we would represent this by having a starting point, then we would draw two branches from that starting point: one for heads and one for tails.

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Tree diagram for first coin flip.
first set of coin flip tree diagram


Now, when we flip the second coin, it can land on either heads or tails. Thus, we could get a heads on the first coin, and then we could get either heads or tails on the second coin, or we could get tails on the first coin, and then get either heads or tails on the second coin.

In the tree diagram, we represent this by drawing two branches off of each of our last branches. These branches represent heads or tails on the flip of the second coin.

Tree diagram for second coin flip.

tree diagram for flipping a coin twice


Our tree diagram displays all the possible outcomes of flipping two coins in succession. Each path of branches represents one outcome. From the diagram, we see that we have four possible outcomes:

  • Heads, Heads
  • Heads, Tails
  • Tails, Heads
  • Tails, Tails

The coin toss example is a simple example of a tree diagram. Let’s look at a few more examples to become more comfortable with this mathematical tool.

Examples

In our first example, let’s assume you are trying to put together an outfit to wear to an upcoming event you are planning to attend. You have three pairs of pants to choose from (P1, P2, P3); two tops (T1, T2); and two pairs of shoes (S1, S2).

How many possible outfits can you make with these clothing items?To calculate the number of outfits we have to choose from, we will create a tree diagram. From our starting point, we will first consider our pants choices. We have three pairs of pants to choose from, so we will have three branches going out from our starting point. We know we have two tops to choose from, so from each of the pants branches, we will draw two branches: one for each top. Lastly, we have two pairs of shoes to choose from, so from each of the shirt branches, we will draw two more branches: one for each pair of shoes.

Tree diagram for outfit choices.

Tree diagram for coin toss and ball color outcomes.
tree diagram


We see that we have six paths of branches, so there are six possible outcomes. Our outcomes include:

  • Heads and a red ball
  • Heads and a yellow ball
  • Heads and a green ball
  • Tails and a red ball
  • Tails and a yellow ball
  • Tails and a green ball

Lesson Summary

Like an actual tree, a tree diagram in mathematics branches out and expands. We see that it is an extremely useful tool in general mathematics, probability, and statistics.

By organizing our information in this manner, we can easily identify and count all our possible outcomes. Now that we know what these tree diagrams are and how to create them, we can add them to our mathematical toolbox and use them as needed.

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