Probability Tree Diagram Maker & Tree Diagram Creator
A tree diagram generator for maths and statistics: probability trees with each branch labelled and every outcome multiplied out, plus decision trees and classification trees, for lessons, worksheets and reports.
Tree diagram styles
Each style is shown on a tree of its own: the real numbers split into their subsets, a spinner spun twice, rain and lateness, a driving test with a retake. Click a card to load the prompt behind it.
Tree diagram examples
Real renders of the four starters: two counters drawn without replacement from a bag of 3 red and 2 blue, a screening test with 2% prevalence worked through to P(test positive), all eight outcomes of three coin tosses, and a product launch decision rolled back to its expected value. Click one to load its prompt.
What is a tree diagram?
A tree diagram is a branching diagram that lists every outcome of a sequence of events; in probability it is the probability tree, where each branch carries the probability of one outcome given the branches before it and multiplying along a path gives the probability of that path. Read from left to right, the first fan of branches is the first event (the first draw, toss or test result), each later fan repeats from the end of every earlier branch, and the end of each path is one outcome of the whole experiment. The branches leaving any point add to 1, and so do all the outcomes together. The same picture also serves decision trees, which add choices and payoffs, and hierarchy trees, which only show what belongs under what.
What this tree diagram maker works out
- The products are computed, not guessed. Each outcome is multiplied out in code along its path and written as the product, the unsimplified fraction and the simplified one, for example 3/5 × 2/4 = 6/20 = 3/10.
- Fractions stay fractions. Branches written as fractions keep their denominators, so the 2/4 of a second draw is not quietly turned into 1/2; decimals and percentages are multiplied as decimals and percentages.
- A missing complement is filled in. When one branch at a point has no probability it gets 1 minus the others (No rain after Rain 0.3 becomes 0.7). Branches you give that add to more or less than 1 are drawn exactly as written, never corrected behind your back.
- Events you name are summed. Ask for P(exactly one red), at least two heads, same color or a single result such as test positive, and the matching outcomes are added and written under the tree.
- Sized to stay readable. Up to four stages and 48 outcomes in one tree: two dice (36 outcomes) fit, three dice (216) do not.
The tree diagrams people make, and what each one needs
Most requests are one of these six. They differ in what goes on the branches and what is written at the ends.
Probability tree diagram
The two- or three-stage tree of school and college statistics: one fan of branches per event, a probability on every branch and the outcome column on the right with each path multiplied out. Give the probabilities as you have them (3/5, 0.6 or 60%) and name the events to add up, such as P(exactly one red).
Tree diagram without replacement
Taking counters, cards or sweets without putting them back changes the second fan: from 3 red and 2 blue, Red 3/5 is followed by Red 2/4 and Blue 2/4, and Blue 2/5 by Red 3/4 and Blue 1/4. Describe the bag and the number of draws and the later fractions are worked out in code. With replacement every stage simply repeats the first.
Conditional probability tree diagram (Bayes)
Screening and quality-control questions: the first branches split on the condition (its prevalence), the second on the test result (sensitivity under the condition, false positive rate without it). With 2% prevalence, 90% sensitivity and 5% false positives, P(test positive) = 1.8% + 4.9% = 6.7%, so only about 27% of positives have the disease.
Coin and dice tree diagrams (sample space)
Every branch equally likely: 1/2 for a fair coin, 1/6 for a die. Three tosses give 8 outcomes of 1/8 each and two dice give 36 of 1/36; ask for an event such as at least two heads (4/8 = 1/2) and it is counted off the tree, the clearest way to show a sample space.
Decision tree diagram
Squares for decisions, circles for chance events and a payoff at the end of every branch. Expected values are rolled back from the payoffs: a launch that succeeds with probability 0.6 for $500,000 and fails for −$200,000 is worth 0.6 × 500,000 + 0.4 × (−200,000) = $220,000, so it beats not launching, and the rejected option is marked off with a double stroke.
Hierarchy and classification tree
No probabilities, only what sits under what: the real numbers split into rational and irrational, the rational into integers and non-integer fractions, and on down. Drawn top to bottom in boxes unless you ask for left to right, with up to four levels below the top box.
The multiplication and addition rules behind every tree
Multiply along a path to get the probability of that path, then add the paths that make up an event. The first rule is the general multiplication rule: every branch after the first stage is a conditional probability, the chance of B given that A has already happened. The second is the addition rule for mutually exclusive outcomes: two different paths of one tree can never both happen, so their probabilities simply add.
P(A and B) = P(A) × P(B | A) P(event) = Σ P(path)
In words
probability of a path = product of its branches; probability of an event = sum of its paths
Bayes' theorem comes off the same tree: P(A | B) = P(A and B) / P(B), the one path through A and B divided by the sum of every path that ends in B. In the screening example that is 1.8% / 6.7% ≈ 26.9%.
Worked outcome table: two counters from 3 red and 2 blue
The outcome column of the first example, laid out as a table. These are the numbers the tree writes at the end of each path.
| Path | Branches multiplied | Probability |
|---|---|---|
| Red, Red | 3/5 × 2/4 | 6/20 = 3/10 |
| Red, Blue | 3/5 × 2/4 | 6/20 = 3/10 |
| Blue, Red | 2/5 × 3/4 | 6/20 = 3/10 |
| Blue, Blue | 2/5 × 1/4 | 2/20 = 1/10 |
| All four paths | 20/20 = 1 |
P(exactly one red) = 6/20 + 6/20 = 12/20 = 3/5. With replacement the second fractions would be 3/5 and 2/5 again, and P(Red, Red) would be 9/25 instead of 3/10.
How to make a tree diagram
List the stages and their outcomes
Write the events in order (first draw, second draw) and the outcomes each one can have. Every outcome of a stage gets its own branch from the end of every branch of the stage before it.
Put a probability on every branch
Use the probability given what has already happened: after a red is taken out and not replaced, the bag holds one red fewer. The branches leaving any one point add to 1, the quickest check that nothing is missing.
Multiply along, then add down
At the end of each path write the product of its branches; over the whole tree those products add to 1. For an event, add the paths that belong to it and nothing else.
Tree diagram vs two-way table vs Venn diagram
A tree diagram shows events in order with the conditional probabilities on its branches; a two-way table shows the same joint probabilities as a grid of cells; a Venn diagram shows overlaps between events but no order.
| Tree diagram | Two-way table | Venn diagram | |
|---|---|---|---|
| Order of events | Shown, stage by stage | Not shown | Not shown |
| Conditional probabilities | Written on the later branches | Worked out from a row or column total | Not shown directly |
| Joint probabilities | At the end of each path | In each cell | In each region |
| More than two stages | One more fan per stage | Awkward beyond two variables | Awkward beyond three sets |
| Best for | Draws, tosses and tests in sequence | Survey counts split by two categories | Questions about A or B, A and B |
Most textbook problems can be drawn all three ways. Reach for the tree when the question is conditional or the events happen one after another.
Tree diagram questions, answered
What is a tree diagram in math and statistics?
It is a diagram that lists every outcome of a multi-step experiment as a path of branches, used to find the probability of combined events. Each fan of branches is one step; the number of paths is the number of outcomes (2 × 2 × 2 = 8 for three coins), and multiplying the probabilities along a path gives that outcome's probability. Counting problems use the same tree without the probabilities.
How do you make a tree diagram for a probability problem?
Draw one fan of branches per event, label each branch with its outcome and probability, then multiply along each path. Here you can type the problem as a sentence (a bag of 3 red and 2 blue, two draws without replacement) or type the branches as an indented list, one per line, with 3/5, 0.6 or 60% after each label; indent a line further to hang it from the line above.
How do you read the probabilities at the end of a tree diagram?
Each number at the end of a path is a joint probability: the chance that every event on that path happens, in that order. P(Red, Blue) = 3/5 × 2/4 = 3/10 means red first and blue second. To answer a question about an event, find every path that fits it and add their end values; all the end values together add to 1.
How does a tree diagram change without replacement?
Without replacement the second fan uses what is left after the first draw, so its fractions differ from branch to branch. From 3 red and 2 blue, the branches after a red are Red 2/4 and Blue 2/4, and after a blue Red 3/4 and Blue 1/4. With replacement every fan repeats 3/5 and 2/5, and the draws are independent.
Why must the branches from each point add up to 1?
Because the branches leaving one point are every possible result of that step, and exactly one of them happens. If the branches after Rain read Late 0.4 and On time 0.5, a result is missing or a number is wrong. This tool fills a single missing branch with the complement and draws any set that does not add to 1 exactly as you wrote it, without correcting it.
How do you find a conditional probability from a tree diagram?
Divide the path you want by the total of all paths that share the condition you are given. For the screening test, P(disease | test positive) = P(disease and positive) / P(test positive) = 1.8% / (1.8% + 4.9%) ≈ 26.9%. This is Bayes' theorem read off the tree, and it is why a positive result from a test for a rare condition is often a false alarm.
Should the fractions on a tree diagram be simplified?
The branches are usually left as written, and the end values are often shown both unsimplified and simplified. Keeping 6/20 next to 3/10 shows the common denominator, so the four outcomes visibly add to 20/20. Decimals and percentages work too; mix them only if your course does, because a tree in one form is easier to check.
Is a tree diagram the same as a family tree, a phylogenetic tree, a cladogram or a dichotomous key?
No: those four are different figures with their own conventions, and each has its own tool here. A family tree of inherited traits is a pedigree chart, an evolutionary tree with branch lengths is a phylogenetic tree, one built from shared derived traits is a cladogram, and an identification guide of paired choices is a dichotomous key. A tree diagram in maths is about outcomes and their probabilities.
Will the AI change the probabilities I give it?
No. Every branch value, product, sum and expected value is worked out in code before the image is drawn, and the model is told each one word for word. It is never asked to multiply 3/5 by 2/4. If your branches do not add to 1, the tree is drawn as you wrote it rather than quietly fixed, so check the numbers you type.
How should a tree diagram be captioned in a report or paper?
Say what the stages are, whether draws were replaced, and which numbers are conditional. For example: Figure 1. Probability tree for two draws without replacement from a bag of 3 red and 2 blue counters. Values on the branches are conditional probabilities; values at the ends are joint probabilities. For a decision tree, also state the payoff units and how expected values were rolled back.
Can an AI-drawn decision tree go into a journal article?
Often yes, because it is an explanatory image rather than a picture of your data, but the journal decides. Elsevier's generative AI policy for journals names decision trees among the explanatory images AI tools may help create, while images of primary research data may not be generated. Check your journal's own policy and say in the methods or legend how the figure was made.
How many outcomes can one tree diagram hold?
Up to four stages and 48 outcomes here, which covers two dice (36 outcomes) and four coin tosses (16). Past that a tree stops being readable on a page: use the counting principle or a table, or draw only the branches the question needs (for example, only the paths with at least one six).
Sources
- 3.5 Tree and Venn Diagrams — OpenStax Introductory Statistics 2e
- Introduction to Probability (Grinstead and Snell), tree diagrams and conditional probability — Dartmouth College
- Generative AI policies for journals — Elsevier
Keep exploring
Our other diagram tools, and the guides behind them.
Draw your tree diagram
From a probability problem to a textbook-clean tree with every outcome worked out, in about a minute.
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