What is the big-O worst-case runtime to find a node in a BST where the height of the tree is h?

Question

BlackTom AI · Accepted Answer

Searching a binary search tree proceeds by repeatedly comparing the target key with the current node and moving one level down (left or right). Each comparison advances the search by one edge, so the cost grows with the number of levels traversed.

1) Relevant knowledge points and concepts
- Binary Search Tree (BST): for each node, left subtree keys < node key < right subtree keys.
- Height h of the tree: the length of the longest root-to-leaf path (measured in edges or levels; either convention changes only constant factors).
- Worst-case runtime: maximum number of node inspections/comparisons needed when searching for a key (present or absent).
- Big-O measures asymptotic upper bound; constants and additive terms are ignored.

2) Step-by-step reasoning
- A search starts at the root and at each step compares the target to the current node’s key, then moves to one child.
- In the worst case the search follows a path from the root to the deepest node (or fails after exploring a path as deep as possible).
- The worst-case number of node visits is proportional to the height h (precisely h+1 node visits if height counts edges and you visit root plus h descendants).

3) Intermediate counting (simple calculation)
- If height h is the maximum number of edges from root to a leaf, number of comparisons ≤ h + 1.
- Big-O notation drops constant +1, so runtime = O(h).

4) Why O(h) is correct
- Each step descends exactly one level; therefore cost scales linearly with depth reached. The deepest possible depth is h, so worst-case cost is Theta(h), and thus O(h).

5) Why other options are incorrect
- O(log h): mixes logarithms incorrectly; typical balanced BST searches cost O(log n) in terms of nodes n, but log(h) is much smaller and not the right relation.
- O(1): implies constant-time search regardless of tree height, which is false except for trivial trees.
- “The cost varies... no worst-case value”: incorrect because the worst-case is well-defined (path of length h).

6) Conclusion
The worst-case runtime to find a node in a BST of height h is O(h), which matches the provided answer.

What is the big-O worst-case runtime to find a node in a BST where the height of the tree is h?单项选择题

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