Consider a binary tree T $=($V$,$ E$)$ where each edge e in E has an associated length l$\_$e$,$ and each vertex v in V holds a quantity of goods g$\_$v $.$ Your task is to select k $+1<=|$V $|$ vertices, including the root, to serve as distribution centers that minimizes the total transport distance of goods from all vertices to their nearest warehouse. The goods can only move in the upward direction towards the root. Give an O$(|$V$|^2$k$^2)$ time algorithm for computing the minimum total transport distance of such a plan.