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.

Let me provide a small example for illustration: Let us have a graph with vertices A$,$ B$,$ C$.$ A is the root of the tree. There is an edge in between A and B with a length l$\_$e of $2.$ There is an edge in between A and C with a length l$\_$e of $3.$ B has a g$\_$v value of $10$ and C has a g$\_$v value of $15.$ A has a g$\_$v value of $0.$ Now let's say that k $=0.$ The root must always be a warehouse. The total cost of moving goods will be $2(10)+3(15)=65.$ Now let's say that k$=1,$ and we place a warehouse at the right child. The total cost of moving goods will be $2(10)=20.$ Now can you redesign your solution based on this example?

Design a DP algorithm for this problem. Give an O$($V$^2$k$^2)$ time algorithm for computing the minimum total transport distance of such a plan.