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## Six Sigma Green Belt–Management Tool #2: Tree Diagram

A tree diagram is used to communicate logical relationships between critical events (such as with failure tree analysis) or specific objectives (decision tree analysis). One event may cause another, and that new event may cause a series of others, so you have a hierarchical relationship that resembles the branching of a tree.

Let’s say you are putting on a company picnic, and you pick a weekend date for it. You are holding a raffle for charity that costs \$5 per ticket and you want to calculate how much money you think you will make on the raffle. You have to have some allowance for if it rains. Let’s say the long-term weather forecast is for it to have an 80% chance of clear skies or scattered clouds, and 20% chance of rain. If you are trying to forecast how much money you will make, then you have to account for both probabilities, whether it will rain or not rain.

Now if it is does not rain, your past experience tells you that there will be 100 people that show up. If it does rain, again your past experience tells you only half the people will show up, giving you only 50 people that will come to the event.

In either case, your past experience tells you that each person on average buys 2 tickets that cost \$5 each, so each person spends \$10 on the raffle on average.

Now what is the expected payout for the event? In the case of “No rain”, there will be 100 people X \$10 spent per person or \$1000. In the case of “Rain”, there will be 50 people X \$10 spent per person or \$500. However since the probability of “No rain” is 80%, and the probability of “Rain is” is 20%, the expected payout will be on average

(Probability of “No rain”) X (Payout of “No rain”) +

(Probability of “Rain”) X (Payout of “Rain”) = (80% X \$1000) + (20% X \$500) = \$800 + \$100 = \$900.

It can be used in decision making, where the decision tree consists of 3 types of nodes:-

1. Decision nodes – commonly represented by squares

2. Chance nodes – represented by circles

3. End nodes – represented by triangles

In the example above, whether it rains or not rains would be represented in squares, the probabilities of 80% and 20% would be represented by the circles that lead from these squares, and the end result we’ve listed above (\$800 payout for “No rain” and \$100 payout for “Rain”) would be represented by triangles at the end of the branches.

It can be used in fault tree analysis, where instead of decisions, you have critical events that occur in the failure of the part. Each of these will have events that cause these failures, and these events will have their causes and so on, where eventually you get to the root causes.

In project management, breaking down a project into activities through the work breakdown structure could also be considered an example of a tree diagram.

In conclusion, tree diagrams can help visualize logical, hierarchical relationships between events, objectives, or tasks. In certain applications, it helps identify the probabilities of the various outcomes, which helps in the calculation of the relative contribution each “branch” makes.