# Optimal Cut

A cut of a binary tree is a set of tree nodes such as for each possible path from the root node to any leaf node, just one node in the cut belongs to the path. In a weighted binary tree, one can compute a weighted cut, because it would include weighted nodes. The total weight of such cut can be computed by adding up the weights of the nodes in it. The figure below shows a weighted binary tree. The gray nodes represent a weighted cut, and its total weight is 14. Now, given a weighted binary tree, you have to write a program that finds the weighted cut with the maximal total weight value among all possible weighted cuts in the tree. This is hereafter called the Optimal Cut of the input binary tree. However, to make it a bit more interesting, your program must find the optimal cut that includes no more than K nodes, and report its weight. In the figure above, for instance, the nodes of the optimal cut sums 28 when K = 3, and 15 when K = 2. Input The input can contain several problems. Each problem contains the description of a complete binary tree in a single line. This description corresponds to sequence of integer numbers, separated of each other by a blank space. The description starts with an integer 0 ≤ H < 20 representing the height of the tree, followed by another integer 1 ≤ K ≤ 20 representing the maximum number of nodes in the optimal cut to be found. Then, the weights −103 ≤ Wi ≤ 103 of the nodes follow, listed in pre-order. The end of input is indicated by a single line containing only an integer value of ‘-1’. Output For each problem in the input, the output should be a single line with a single integer number, repre- senting the total weight of the optimal cut found. Sample Input 2 3 8 6 7 -2 -1 2 1 011 3 3 -8 1 0 0 1 2 1 1 -1 1 1 1 3 1 4 -1 Sample Output 9 1 5