Solve transaction aggregation and search tasks

# Solution Alignment The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests. ## Problem Restatement Complete the following coding tasks: (a) Given a list of transactions, each with userId and amount (credits positive, debits negative), compute the final balance for each userId. Return a map from userId to balance. Handle large inputs efficiently and specify time/space complexity. (b) Given an array of positive integers nums and a list of targets, for each target find the minimal prefix length k such that sum(nums[0..k-1]) >= target; return -1 if no such k exists. Implement using a prefix-sum array and binary search, and analyze complexity. (c) For an immutable integer array and a small number of range-sum queries [L, R] with no updates, write a simple recursive divide-and-conquer function ... ## Recommended Approach Choose traversal based on the required view or aggregate. DFS is natural for subtree computations and reconstruction; BFS is natural for level order or side views. Keep per-depth or per-position state when the output depends on columns, rows, or depths. ## Correctness The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call. ## Complexity Most tree traversals are O(n) time and O(h) recursion stack for DFS or O(w) queue space for BFS, where h is height and w is maximum width. ## Edge Cases and Tests Empty tree, one node, skewed tree, duplicate values when reconstruction assumes uniqueness, deep recursion, and tie-breaking for same row/column nodes.