Given the root of a binary tree and a target sum, count the number of downward paths whose values add up to the target, where paths do not need to start at the root or end at a leaf.

"I went through four rounds: a 90-minute OA, a live coding plus behavioral round, a tree plus Leadership Principles round, and then a hiring manager final with light system design. The OA was two coding questions, then the live rounds stacked coding with behavioral in a pretty straightforward way. Most of it landed around medium difficulty for me. The round that felt hardest was the tree plus Leadership Principles one because that was where the optimization follow-ups and deeper probing really picked up. The final round was more of a high-level package-tracking design discussion than a super deep system design."