Transform a binary tree so that the original left-most leaf becomes the new root. The original root becomes a left child, and the original right child becomes the right child of the original root's parent.

Design a data structure that supports insertion, deletion, and random element retrieval, all in constant time on average. You'll need to handle edge cases carefully to achieve O(1) complexity.

You're given a nested list of integers where each integer has a depth. Calculate the weighted sum of all integers, where the weight is the depth of the integer from the bottom of the nested structure.