# 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 Solve the following coding tasks: 1) In a sorted array, every value appears exactly twice except for one value that appears once. Find the single value in O(log n) time and O( 1) extra space using a binary-search-based approach. 2) Given the root of an N-ary tree and a sequence of integers representing a path, determine whether this path exists in the tree. Each node may have multiple children and values are not guaranteed to be unique. A valid path must start from the root but does not need to end at a leaf node. Return true/false and analyze time and space complexity. 3) Given two integer arrays, compute the maximum length of a contiguous subarray that appears in both arrays. Discuss multi... ## 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.