Final answer:
The worst case time complexity for inserting a new key in a binary search tree containing N keys is O(log N).
Step-by-step explanation:
The correct answer is a) O(log N). When inserting a new key into a binary search tree containing N keys, the worst-case time complexity is O(log N). This is because the binary search tree has a hierarchical structure that allows for efficient search and insertion operations. The height of a balanced binary search tree is log N, and since the worst-case time complexity is determined by the height of the tree, it is O(log N). For example, if a binary search tree has 16 nodes, the maximum height of the tree is log2(16) = 4, indicating that it will take at most 4 comparisons to find the correct position for a new key during insertion.