Group a tree by vertical columns

# 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 Given the root of a binary tree, produce a vertical listing of nodes: assign each node an x-coordinate (root at x=0; left child x−1; right child x+ 1). For each x from smallest to largest, list node values from top to bottom; when multiple nodes share the same (x, depth), order them by their left-to-right appearance in level order. Return a list of columns (each a list of values). Describe your traversal strategy, tie-breaking, and data structures. Analyze time and space complexity. ## Recommended Approach Choose traversal based on the required output. DFS is natural for subtree computations, reconstruction, and range pruning; 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.