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Tree Diagrams in Math: Definition & Examples

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  • 0:03 Tree Diagrams
  • 1:53 Examples
  • 4:12 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

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.


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?

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