Read an Expression as a Tree, Not a Sentence — × Binds Tighter Than +, Full Stop

You were taught to read 2 + 3 \times 4 by scanning left-to-right and applying BODMAS: "multiplication before addition, so do 3 \times 4 first, then add 2." That works, but it requires you to override the left-to-right reading with a rule you memorised. The better mental model is to stop reading the expression as a sentence entirely. Read it as a tree.

The intuition

Every arithmetic expression has a hidden skeleton: a tree where each operation is a node, and the node's children are the things it acts on. Multiplication nodes sit deeper in the tree than addition nodes, because multiplication grabs its operands tighter. When you evaluate, you collapse the tree from the bottom up — inner branches first, outer branches last. BODMAS is not a priority list you apply externally; it is the shape of the tree, and the tree is what the expression secretly is all along.

Once you see the tree, you stop doing arithmetic in your head by chanting "BODMAS" and start doing it by evaluating leaves first and working outward. The rule becomes automatic because it is structural, not mnemonic.

The expression $2 + 3 \times 4$ as a tree. The $\times$ node is deeper than the $+$ node, because multiplication binds tighter — it grabs its two leaves first and collapses to $12$. Only then does the outer $+$ see anything; by that point one of its children is already the single value $12$, and $2 + 12 = 14$ finishes the job. You never had to "remember BODMAS" — you just evaluated leaves first.

Why the tree shape is not arbitrary

Multiplication binding tighter than addition is not a random rule someone invented — it mirrors how multiplication means repeated addition. Writing 2 + 3 \times 4 expands in your head to 2 + (4 + 4 + 4). The 3 \times 4 has to happen first because it is the three 4s being added, and those three 4s are themselves a single quantity before the outer +2 joins in. If addition bound tighter, 2 + 3 \times 4 would mean (2 + 3) + (2 + 3) + (2 + 3) + (2 + 3) = 20 — and that does not match the meaning of 2 plus "three fours." The tree shape reflects what the expression is trying to say.

The same structural intuition works for every other binary operator.

Exponents bind tighter than multiplication. 2 \times 3^2 = 2 \times 9 = 18, not (2 \times 3)^2 = 36. The 3^2 is one number before the 2 multiplies it.
Multiplication binds tighter than addition. Already seen.
Function application binds tightest of all. 2 + \sin(x) is 2 + \sin(x), not \sin(2 + x). The function gobbles its argument first.

Try it: watch the tree collapse

Drag $x$. The correct tree-reading of $2 + 3x$ groups $3$ and $x$ first, then adds $2$. The wrong flat reading would group $2$ and $3$ first — giving $(2+3)x = 5x$. The two answers coincide only at $x = 0$; everywhere else the gap is exactly $2(1 - x)$, which is *never* zero for $x \neq 1$. Tree structure is not optional.

How to build the tree by eye

When you look at a long expression, the trick is:

Find the outermost operator — the one that would be evaluated last if you followed BODMAS. That is your root.
The root's children are the sub-expressions on each side of that operator.
Recurse into each sub-expression.

For 2 + 3 \times 4, the outermost operator is + (addition is weakest and binds loosest). So + is the root, with children 2 and 3 \times 4. Recurse into 3 \times 4: its root is \times, children 3 and 4. Done — you have the tree.

For a messier expression like 5 + 2 \times (x - 1)^2:

Outermost is + — root has children 5 and 2 \times (x-1)^2.
Inside the right child: outermost is \times — its children are 2 and (x-1)^2.
Inside (x-1)^2: outermost is ^2 (exponentiation) — its children are (x-1) and 2 (the exponent).
Inside (x - 1): outermost is - — children are x and 1.

Four levels deep. When you evaluate, you collapse from the deepest leaves upward.

What you get from the tree view

BODMAS stops being a memory task. The order falls out of the tree shape, and the tree shape falls out of what the symbols mean.
Algebra errors become visible. When a student writes (a + b)^2 = a^2 + b^2, the mistake is that they collapsed a non-leaf node too early. Looking at the tree — ^2 sitting above a + — you can see immediately that the exponent acts on the whole (a + b) subtree, not on each leaf individually.
Coding parsers becomes trivial. Every programming language that evaluates expressions builds a tree like this (an abstract syntax tree). You are learning, for free, the same mental model that compilers use.
Reading math becomes faster. Long expressions no longer look intimidating. You decompose into subtrees, solve each, plug back in.

A final gut-check

Next time you see 2 + 3 \times 4, don't think "BODMAS says multiplication first." Think: "3 and 4 are married to each other by the \times; they're a single number already (12) before the outer + even looks at them." That is the tree speaking. Multiplication binds tighter than addition, full stop — because that is what the tree shape means.