How to determine the time complexity of search or insert operations in a binary tree?

The time complexity of search and insert operations in a binary tree depends on its height.

• In the worst case (when the tree degenerates into a list), the height equals the number of elements n, and the operations have a complexity of O(n).
• In the average and best case, if the tree is balanced, the height is O(log n), and the operations are performed in O(log n).

To estimate the complexity, the structure of the tree must be considered:

• Search: starts from the root, at each step choosing the left or right child, reducing the search space by half.
• Insert: similar to search, to find the place for the new node.

If the tree is balanced (for example, AVL or red-black tree), the operations are efficient and have logarithmic complexity.