D. Paths on the Treetime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a rooted tree consisting of n vertices. The vertices are numbered from 1 to n, and the root is the vertex 1. You are also given a score array s_1, s_2, \ldots, s_n.A multiset of k simple paths is called valid if the following two conditions are both true.   Each path starts from 1.  Let c_i be the number of paths covering vertex i. For each pair of vertices (u,v) (2\le u,v\le n) that have the same parent, |c_u-c_v|\le 1 holds.  The value of the path multiset is defined as \sum\limits_{i=1}^n c_i s_i.It can be shown that it is always possible to find at least one valid multiset. Find the maximum value among all valid multisets.InputEach test contains multiple test cases. The first line contains a single integer t (1 \leq t \leq 10^4) — the number of test cases. The description of the test cases follows.The first line of each test case contains two space-separated integers n (2 \le n \le 2 \cdot 10^5) and k (1 \le k \le 10^9) — the size of the tree and the required number of paths.The second line contains n - 1 space-separated integers p_2,p_3,\ldots,p_n (1\le p_i\le n), where p_i is the parent of the i-th vertex. It is guaranteed that this value describe a valid tree with root 1.The third line contains n space-separated integers s_1,s_2,\ldots,s_n (0 \le s_i \le 10^4) — the scores of the vertices.It is guaranteed that the sum of n over all test cases does not exceed 2 \cdot 10 ^ 5.OutputFor each test case, print a single integer — the maximum value of a path multiset.ExampleInput
25 41 2 1 36 2 1 5 75 31 2 1 36 6 1 4 10Output
54
56
NoteIn the first test case, one of optimal solutions is four paths 1 \to 2 \to 3 \to 5, 1 \to 2 \to 3 \to 5, 1 \to 4, 1 \to 4, here c=[4,2,2,2,2]. The value equals to 4\cdot 6+ 2\cdot 2+2\cdot 1+2\cdot 5+2\cdot 7=54.In the second test case, one of optimal solution is three paths 1 \to 2 \to 3 \to 5, 1 \to 2 \to 3 \to 5, 1 \to 4, here c=[3,2,2,1,2]. The value equals to 3\cdot 6+ 2\cdot 6+2\cdot 1+1\cdot 4+2\cdot 10=56.
25 41 2 1 36 2 1 5 75 31 2 1 36 6 1 4 10
54 56 
['dfs and similar', 'dp', 'greedy', 'sortings', 'trees', '*1900']