A tree diagram is a diagram using branches from locations (like a tree) to show possibilities and relationships including:

• all possible outcomes of an experiment
• all possible arrangements of a set of elements
• match-ups in a tournament
• prime factorization of a natural number
• etc.

Experiment: Toss a penny and a die.

Two possible tree diagrams illustrating every possible outcome are are shown below.

Either way of looking at the set of outcomes result in twelve possibilities.

Set of elements: Math class (M), English class (E), Physical Education (P) class and a spare (S).

One possible tree diagram of all possible arrangements of the four classes is shown below.

There are twenty-four possibile arrangements of the four classes.

In a tournament consisting eight teams, one possible way of arranging which team plays which other team is shown in the tree diagram below.

The number 40 can be written using prime factorization as 2 x 2 x 2 x 5. All of the factors are prime numbers.

Using exponential form, 40 = 2³5¹, indicating that there are three 2's and one 5 multilplied together to get the result of 40.

A different tree diagram will occur, depending on how the number (40) is factored, but each different diagram will yield the same result. Some examples are given below.