Fenwick Tree: Mental Model

Start with the idea

Begin with the concrete question: Apply the 1-indexed add operations and return the prefix sum through query. For the visible input n=5 adds=(1,5),(3,2),(4,4) query=3, the stated result is 7. Before reading code, name the smallest piece of information that must survive from one step to the next. A range method summarizes intervals so the same information is not recomputed. A static prefix table and an updateable tree make different tradeoffs; define endpoint conventions first. That is the mental model for Fenwick Tree; it gives you a reason for each update instead of a line of code to memorise.

The core mechanism is this: Store partial sums in power-of-two ranges and move indexes by their lowest set bit. Treat it as a claim to test after every operation. Write down the state before the first operation, update it once, and ask which part of the claim is still true. If an item is skipped, inserted, or removed, explain why that action cannot destroy an answer that should be retained. A useful trace has columns for the current input position, the state before the step, the action, and the state afterward. Use the visible sample until the expected result becomes inevitable rather than merely plausible.

Now inspect the Rust sketch, not just its final expression. The supplied Rust sketch is a starting point.