Final answer:
Inserting an item in a self-balancing BST like an AVL or Red-Black tree has a logarithmic time complexity, O(log n), in both average and worst-case scenarios due to the tree's self-balancing properties that maintain its height at approximately log n.
Step-by-step explanation:
The operation of inserting an arbitrary item in a self-balancing binary search tree (BST), such as an AVL tree or a Red-Black tree, has a time complexity that can be evaluated in terms of its average and worst-case scenarios. In the average case, the insertion operation can be considered to have logarithmic growth, or O(log n), where 'n' is the number of nodes in the tree. This efficiency is due to the self-balancing nature of these trees which ensures that the height of the tree is approximately log n, thereby allowing operations to run quickly.
In the worst-case scenario, the growth rate of insertion still remains O(log n) for a self-balancing BST. This is because operations such as rotations in AVL trees or color flips and rotations in Red-Black trees are used to restore balance after an insertion, keeping the height of the tree balanced and preventing the degradation into a list-like structure, which would have a linear growth rate.