A frequency tree sorts a population by two successive categories, using actual counts at every node (always whole numbers) — unlike a probability tree, which uses fractions or decimals along the branches. At any point in the tree, the numbers on the branches leading out of a node must add up to the number entering it.

Place every value you're given directly, then use subtraction wherever a branch pair has one known and one unknown value — each node's total minus its known branch gives the missing branch.

Example: 30 students took a test; 23 passed. Number who failed: $30-23=7$.

Keep working outward branch by branch, and as a final check, confirm that all the terminal (end) branches add back up to the overall total.

Convert it to an integer immediately, before putting anything on the tree — never carry a percentage or fraction through the subtraction steps.

Example: $120$ people in total, two-fifths of whom are female: $\frac{2}{5}\times120=48$ females, so $120-48=72$ males.

For a large population, it's often faster to calculate as many of the first-level frequencies as possible directly by multiplication (rather than leaving several to be found later by subtraction) — this reduces the number of places an earlier slip can cascade into a later answer.

Example: In a tree with total $N=40$ and a terminal node of $14$ for "Male, likes coffee," $P(\text{Male and coffee}) = \dfrac{14}{40} = \dfrac{7}{20}$.

The "given" group becomes the new, restricted total — not the full population $N$.

Example: Of 80 people, 53 are left-handed, of whom 27 are female. Find $P(\text{female}\mid\text{left-handed})$.

A survey of 200 film fans found that 40% prefer action films. Of those who prefer action films, 70 are male. Of those who do not prefer action films, 50 are female.

1. Convert the percentage: $200\times0.40=80$ prefer action; $200-80=120$ do not.
2. Action branch: $70$ male, so $80-70=10$ female.
3. Non-action branch: $50$ female, so $120-50=70$ male.
4. Check: terminal nodes sum to $70+10+50+70=200$ ✓.
5. Find $P(\text{male}\mid\text{prefers action})$: $\dfrac{70}{80}=\dfrac{7}{8}$.