#D8791. Path Queries
Path Queries
Path Queries
You are given a weighted tree consisting of n vertices. Recall that a tree is a connected graph without cycles. Vertices u_i and v_i are connected by an edge with weight w_i.
You are given m queries. The i-th query is given as an integer q_i. In this query you need to calculate the number of pairs of vertices (u, v) (u < v) such that the maximum weight of an edge on a simple path between u and v doesn't exceed q_i.
Input
The first line of the input contains two integers n and m (1 ≤ n, m ≤ 2 ⋅ 10^5) — the number of vertices in the tree and the number of queries.
Each of the next n - 1 lines describes an edge of the tree. Edge i is denoted by three integers u_i, v_i and w_i — the labels of vertices it connects (1 ≤ u_i, v_i ≤ n, u_i ≠ v_i) and the weight of the edge (1 ≤ w_i ≤ 2 ⋅ 10^5). It is guaranteed that the given edges form a tree.
The last line of the input contains m integers q_1, q_2, ..., q_m (1 ≤ q_i ≤ 2 ⋅ 10^5), where q_i is the maximum weight of an edge in the i-th query.
Output
Print m integers — the answers to the queries. The i-th value should be equal to the number of pairs of vertices (u, v) (u < v) such that the maximum weight of an edge on a simple path between u and v doesn't exceed q_i.
Queries are numbered from 1 to m in the order of the input.
Examples
Input
7 5 1 2 1 3 2 3 2 4 1 4 5 2 5 7 4 3 6 2 5 2 3 4 1
Output
21 7 15 21 3
Input
1 2 1 2
Output
0 0
Input
3 3 1 2 1 2 3 2 1 3 2
Output
1 3 3
Note
The picture shows the tree from the first example:
inputFormat
Input
The first line of the input contains two integers n and m (1 ≤ n, m ≤ 2 ⋅ 10^5) — the number of vertices in the tree and the number of queries.
Each of the next n - 1 lines describes an edge of the tree. Edge i is denoted by three integers u_i, v_i and w_i — the labels of vertices it connects (1 ≤ u_i, v_i ≤ n, u_i ≠ v_i) and the weight of the edge (1 ≤ w_i ≤ 2 ⋅ 10^5). It is guaranteed that the given edges form a tree.
The last line of the input contains m integers q_1, q_2, ..., q_m (1 ≤ q_i ≤ 2 ⋅ 10^5), where q_i is the maximum weight of an edge in the i-th query.
outputFormat
Output
Print m integers — the answers to the queries. The i-th value should be equal to the number of pairs of vertices (u, v) (u < v) such that the maximum weight of an edge on a simple path between u and v doesn't exceed q_i.
Queries are numbered from 1 to m in the order of the input.
Examples
Input
7 5 1 2 1 3 2 3 2 4 1 4 5 2 5 7 4 3 6 2 5 2 3 4 1
Output
21 7 15 21 3
Input
1 2 1 2
Output
0 0
Input
3 3 1 2 1 2 3 2 1 3 2
Output
1 3 3
Note
The picture shows the tree from the first example:
样例
3 3
1 2 1
2 3 2
1 3 2
1 3 3