Heavy-Light Decomposition: Mental Model

Start with the idea

Begin with the concrete question: Return the sum on path 2 to 4. For the visible input edges=(0,1),(1,2),(1,3),(3,4) values=[1,2,3,4,5], the stated result is 14. Before reading code, name the smallest piece of information that must survive from one step to the next. An advanced structure or offline query order packages an invariant across operations. Name what each node, block, or reordered query state stores. That is the mental model for Heavy-Light Decomposition; it gives you a reason for each update instead of a line of code to memorise.

The core mechanism is this: Choose one heavy child per node so any root path crosses only logarithmically many light edges. 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 an invariant note rather than a complete program. Turn it into explicit state and transitions before coding.