To analyze the complexity of binary search trees (BST) in the context of Design and Analysis of Algorithms (DAA), we typically consider two main operations: insertion and search.

1. Insertion Complexity:

When inserting a new element into a BST, we start at the root and compare the element with each node to determine its appropriate position in the tree. The complexity of insertion in a BST depends on the height of the tree.

In the best-case scenario, if the tree is perfectly balanced, the height of the tree is logarithmic with respect to the number of nodes (h = log₂(n)), resulting in an average time complexity of O(log n) for insertion.

However, in the worst-case scenario, if the tree becomes highly unbalanced (resembling a linked list), the height of the tree can be equal to the number of nodes (h = n), resulting in a time complexity of O(n) for insertion.

To ensure balanced BSTs and reduce the chances of worst-case scenarios, various self-balancing techniques, such as AVL trees and Red-Black trees, are used. These techniques guarantee a logarithmic height, resulting in an average complexity of O(log n) for insertion.

2. Search Complexity:

The search operation in a BST involves comparing the target element with the values stored in the nodes and navigating left or right based on the comparison until the target is found or determined to be absent. Similar to insertion, the complexity of search in a BST depends on the height of the tree.

In the best-case scenario, where the tree is balanced, the height is logarithmic (h = log₂(n)), resulting in an average time complexity of O(log n) for search.

In the worst-case scenario, where the tree is highly unbalanced, the height of the tree can be equal to the number of nodes (h = n), resulting in a time complexity of O(n) for search.

Again, self-balancing techniques like AVL trees and Red-Black trees help maintain balance, ensuring a logarithmic height and an average complexity of O(log n) for search.

It's worth noting that the complexities mentioned above assume a balanced BST. If the BST is not balanced, the complexities may degrade to the worst-case scenario.

In conclusion, the complexity of binary search trees depends on their balance. If the tree is balanced or using self-balancing techniques, the average complexities for insertion and search operations are O(log n). However, in the worst-case scenario, when the tree becomes highly unbalanced, the complexities can degrade to O(n).

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