Final answer:
Sorting N items using a vanilla Binary Search Tree (BST) has an average case time complexity of O(N log N) if the tree is balanced; worst case can degrade to O(N²) if the tree is unbalanced.
Step-by-step explanation:
When sorting N items using a vanilla Binary Search Tree (BST), the average case time complexity is O(N log N) assuming the tree is balanced. For each insertion, the average case time complexity is O(log N) because we would expect to traverse a portion of the tree logarithmic to the number of nodes to find the insertion point. Given that we must insert N items, this operation is performed N times, leading to an O(N log N) complexity for the entire sorting process. However, in the worst case, if the tree becomes unbalanced and takes the form of a linked list, the complexity would degrade to O(N²) because each insertion could take linear time relative to the number of items already inserted.