Minimum Increments to Equalize Leaf Paths

Given a rooted tree with node traversal costs, you may increase the cost of any nodes by arbitrary non-negative amounts so that all root-to-leaf path sums become equal; compute the minimum number of nodes that must be increased. The core challenge is choosing nodes (including internal ones whose increases affect multiple descendant leaves) to cover the deficits up to the maximum path sum while minimizing the count of modified nodes.