Recall that weight-balanced trees maintain balance by ensuring that the weight of each subtree is within a constant factor of its sibling's weight. How does this property influence the amortized cost of operations like insertion and deletion?

Consider the potential method for analyzing amortized costs. Define a suitable potential function that captures the "imbalance" in the tree, and derive the amortized cost of a rebalancing operation (e.g., splitting or merging subtrees) using this function.

Prove that the total number of rebalancing operations (e.g., splits or merges) over a sequence of m operations is O(m) by carefully bounding the potential function's growth. How does this relate to the amortized cost per operation?