CF1606F Tree Queries

You are given a tree consisting of $ n $ vertices. Recall that a tree is an undirected connected acyclic graph. The given tree is rooted at the vertex $ 1 $ . You have to process $ q $ queries. In each query, you are given a vertex of the tree $ v $ and an integer $ k $ . To process a query, you may delete any vertices from the tree in any order, except for the root and the vertex $ v $ . When a vertex is deleted, its children become the children of its parent. You have to process a query in such a way that maximizes the value of $ c(v) - m \cdot k $ (where $ c(v) $ is the resulting number of children of the vertex $ v $ , and $ m $ is the number of vertices you have deleted). Print the maximum possible value you can obtain. The queries are independent: the changes you make to the tree while processing a query don't affect the tree in other queries.

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The tree in the first example is shown in the following picture: Answers to the queries are obtained as follows: 1. $ v=1,k=0 $ : you can delete vertices $ 7 $ and $ 3 $ , so the vertex $ 1 $ has $ 5 $ children (vertices $ 2 $ , $ 4 $ , $ 5 $ , $ 6 $ , and $ 8 $ ), and the score is $ 5 - 2 \cdot 0 = 5 $ ; 2. $ v=1,k=2 $ : you can delete the vertex $ 7 $ , so the vertex $ 1 $ has $ 4 $ children (vertices $ 3 $ , $ 4 $ , $ 5 $ , and $ 6 $ ), and the score is $ 4 - 1 \cdot 2 = 2 $ . 3. $ v=1,k=3 $ : you shouldn't delete any vertices, so the vertex $ 1 $ has only one child (vertex $ 7 $ ), and the score is $ 1 - 0 \cdot 3 = 1 $ ; 4. $ v=7,k=1 $ : you can delete the vertex $ 3 $ , so the vertex $ 7 $ has $ 5 $ children (vertices $ 2 $ , $ 4 $ , $ 5 $ , $ 6 $ , and $ 8 $ ), and the score is $ 5 - 1 \cdot 1 = 4 $ ; 5. $ v=5,k=0 $ : no matter what you do, the vertex $ 5 $ will have no children, so the score is $ 0 $ ; 6. $ v=7,k=200000 $ : you shouldn't delete any vertices, so the vertex $ 7 $ has $ 4 $ children (vertices $ 3 $ , $ 4 $ , $ 5 $ , and $ 6 $ ), and the score is $ 4 - 0 \cdot 200000 = 4 $ .