Final answer:
Adding a new key to an unbalanced Binary Search Tree (BST) has a worst-case time complexity of O(N), as the structure may degrade to a linked list where the insertion requires traversing the entire tree.
Step-by-step explanation:
When adding a new key to a vanilla Binary Search Tree (BST) that contains N keys, the worst-case time complexity is O(N). This is because in the worst case, the binary search tree can become unbalanced and resemble a linked list, rather than a tree. In such a case, you might need to traverse the entire height of the tree to find the correct insertion point for the new key, and the height of the tree can be as large as N in this unbalanced scenario. Therefore, the insertion operation of a new key in a BST that has degraded into a linked list-like structure would have a linear relationship with the number of keys in the tree.