Use the BST property to prune branches: if the current node's value is less than the low bound, only the right subtree can contain values in range; if greater than the high bound, only the left subtree. Otherwise, include the current node's value and recursively sum both subtrees. This yields O(n) worst-case but typically O(log n + k) time where k is the number of nodes in range.
Pro tip: Clarify whether the range is inclusive and whether the tree can be empty or have duplicate values. Mention that pruning makes it more efficient than a full traversal, and be prepared to discuss iterative vs recursive trade-offs.
Confirm that the range is inclusive, the tree is a valid BST, and node values are integers. Ask about edge cases like empty tree or range with no matches.
Explain that you'll traverse the tree but skip subtrees that cannot contain values in range based on BST ordering. This avoids unnecessary visits.
Define a helper that takes a node and returns the sum. If node is null, return 0. If node.val < low, recurse right; if node.val > high, recurse left; else return node.val + recurse(left) + recurse(right).
State that time is O(n) worst-case (e.g., all nodes in range) but O(log n + k) on average, where k is the number of nodes in range. Space is O(h) for recursion stack, h being tree height.
Walk through a small BST and a range to verify correctness, including cases where the range excludes entire subtrees. Mention potential follow-ups like iterative solution or handling large trees.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying the problem constraints: tree size, query frequency, update frequency, and query types. Then present two main optimization strategies: augmenting nodes with subtree sums for efficient range queries and updates, and precomputing an in-order array with prefix sums for fast lookups but costly updates. Compare their time complexities and discuss trade-offs based on read-heavy vs. write-heavy workloads, and mention hybrid or advanced structures like Fenwick trees or segment trees if appropriate.
Pro tip: Demonstrate awareness of real-world constraints: in practice, updates are often batched or the tree is static, so the optimal choice depends on the actual workload. Also, mention that augmenting nodes requires careful maintenance during rotations in balanced trees.
Ask about the tree size, frequency of queries and updates, and the exact nature of range queries (e.g., sum over a range of keys). This determines the appropriate data structure.
Explain that each node stores the sum of its subtree. Range queries can be answered by traversing the tree and combining subtree sums, achieving O(log n) time for balanced trees. Updates require updating all ancestors, also O(log n).
Describe flattening the tree into a sorted array and building a prefix sum array. Range sum queries become O(log n) via binary search and prefix sum difference, but updates require rebuilding the array and prefix sums, costing O(n).
For read-heavy workloads, precomputed prefix sums offer faster queries (O(log n) with low constant) but updates are expensive. For write-heavy workloads, augmented trees provide balanced O(log n) for both queries and updates. Discuss hybrid approaches like Fenwick trees or segment trees.
Conclude that if updates are frequent, augmented trees or Fenwick trees are better; if the tree is static or updates are rare, precomputed prefix sums are optimal. Mention that in practice, one might use a balanced BST with subtree sums or a Fenwick tree over the in-order array.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.