**This article provides a step-by-step approach to decision trees, using a simple example to guide you through.**

There is no universal set of symbols used when drawing a decision tree but the most common ones that we tend to come across in accountancy education are squares (□), which are used to represent ‘decisions’ and circles (○), which are used to represent ‘outcomes.’ Therefore, I shall use these symbols in this article and in any suggested solutions for exam questions where decision trees are examined.

### Decision trees and multi-stage decision problems

A decision tree is a diagrammatic representation of a problem and on it we show all possible courses of action that we can take in a particular situation and all possible outcomes for each possible course of action. It is particularly useful where there are a series of decisions to be made and/or several outcomes arising at each stage of the decision-making process. For example, we may be deciding whether to expand our business or not. The decision may be dependent on more than one uncertain variable.

For example, sales may be uncertain but costs may be uncertain too. The value of some variables may also be dependent on the value of other variables too: maybe if sales are 100,000 units, costs are $4 per unit, but if sales are 120,000 units costs fall to $3.80 per unit. Many outcomes may therefore be possible and some outcomes may also be dependent on previous outcomes. Decision trees provide a useful method of breaking down a complex problem into smaller, more manageable pieces.

There are two stages to making decisions using decision trees. The first stage is the construction stage, where the decision tree is drawn and all of the probabilities and financial outcome values are put on the tree. The principles of relevant costing are applied throughout – ie only relevant costs and revenues are considered. The second stage is the evaluation and recommendation stage. Here, the decision is ‘rolled back’ by calculating all the expected values at each of the outcome points and using these to make decisions while working back across the decision tree. A course of action is then recommended for management.

### Constructing the tree

A decision tree is always drawn starting on the left hand side of the page and moving across to the right. Above, I have mentioned decisions and outcomes. Decision points represent the alternative courses of action that are available to you. These are within your control – it is your choice. You either take one course of action or you take another. Outcomes, on the other hand, are not within your control. They are dependent on the external environment – for example, customers, suppliers and the economy. Both decision points and outcome points on a decision tree are always followed by branches. If there are two possible courses of action – for example, there will be two branches coming off the decision point; and if there are two possible outcomes – for example, one good and one bad, there will be two branches coming off the outcome point. It makes sense to say that, given that decision trees facilitate the evaluation of different courses of actions, all decision trees must start with a decision, as represented by a □.

A simple decision tree is shown below. It can be seen from the tree that there are two choices available to the decision maker since there are two branches coming off the decision point. The outcome for one of these choices, shown by the top branch off the decision point, is clearly known with certainty, since there is no outcome point further along this top branch. The lower branch, however, has an outcome point on it, showing that there are two possible outcomes if this choice is made. Then, since each of the subsidiary branches off this outcome point also has a further outcome point on with two branches coming off it, there are clearly two more sets of outcomes for each of these initial outcomes. It could be, for example, that the first two outcomes were showing different income levels if some kind of investment is undertaken and the second set of outcomes are different sets of possible variable costs for each different income level.