Problem:
Score : 800 points

Problem Statement

Snuke found a record of a tree with N vertices in ancient ruins.
The findings are as follows:

The vertices of the tree were numbered 1,2,...,N, and the edges were numbered 1,2,...,N-1.

Edge i connected Vertex a_i and b_i.

The length of each edge was an integer between 1 and 10^{18} (inclusive).

The sum of the shortest distances from Vertex i to Vertex 1,...,N was s_i.

From the information above, restore the length of each edge.
The input guarantees that it is possible to determine the lengths of the edges consistently with the record.
Furthermore, it can be proved that the length of each edge is uniquely determined in such a case.

Constraints

2 \leq N \leq 10^{5}

1 \leq a_i,b_i \leq N

1 \leq s_i \leq 10^{18}

The given graph is a tree.

All input values are integers.

It is possible to consistently restore the lengths of the edges.

In the restored graph, the length of each edge is an integer between 1 and 10^{18} (inclusive).

Partial Scores

In the test set worth 300 points, a_i = i, b_i = i+1.

In the test set worth 200 points, N \geq 3, a_i = 1, b_i = i+1.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
:
a_{N-1} b_{N-1}
s_1 s_2 ... s_{N}

Output

Print N-1 lines. The i-th line must contain the length of Edge i.

Sample Input 1

4
1 2
2 3
3 4
8 6 6 8

Sample Output 1

1
2
1

The given record corresponds to the tree shown below:

Sample Input 2

5
1 2
1 3
1 4
1 5
10 13 16 19 22

Sample Output 2

1
2
3
4

The given record corresponds to the tree shown below:

Sample Input 3

15
9 10
9 15
15 4
4 13
13 2
13 11
2 14
13 6
11 1
1 12
12 3
12 7
2 5
14 8
1154 890 2240 883 2047 2076 1590 1104 1726 1791 1091 1226 841 1000 901

Sample Output 3

5
75
2
6
7
50
10
95
9
8
78
28
89
8

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.IO;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using System.Text;
using System.Numerics;
using System.Threading.Tasks;
using System.Text.RegularExpressions;
using static System.Math;
using Debug = System.Diagnostics.Debug;
using LayoutKind = System.Runtime.InteropServices.LayoutKind;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;


static class P
{
    static void Main()
    {
        int n = int.Parse(Console.ReadLine());
        List<int>[] neighbours = Enumerable.Repeat(0, n).Select(_ => new List<int>()).ToArray();
        int[][] edges = new int[n - 1][]; 
        for (int i = 0; i < n - 1; i++)
        {
            var ab = Console.ReadLine().Split().Select(x => int.Parse(x) - 1).ToArray();
            edges[i] = ab;
            neighbours[ab[0]].Add(ab[1]);
            neighbours[ab[1]].Add(ab[0]);
        }
        OmniTreeDP treeDP = new OmniTreeDP(neighbours);
        var sums = Console.ReadLine().Split().Select(long.Parse).ToArray();
        int eqIndex = -1;
        List<long> res = new List<long>();
        for (int i = 0; i < edges.Length; i++)
        {
            var edge = edges[i];
            var atob = treeDP.Query(edge[0], edge[1]);
            var btoa = treeDP.Query(edge[1], edge[0]);
            if (atob == btoa)
            {
                res.Add(0);
                eqIndex = i;
            }
            else
            {
                res.Add(Abs((sums[edge[0]] - sums[edge[1]]) / (atob - btoa)));
            }
        }
        if (eqIndex >= 0)
        {
            BigInteger sum = new BigInteger(0);
            for (int i = 0; i < sums.Length; i++) sum += sums[i];
            for (int i = 0; i < edges.Length; i++)
            {
                var edge = edges[i];
                BigInteger atob = treeDP.Query(edge[0], edge[1]);
                BigInteger btoa = treeDP.Query(edge[1], edge[0]);
                sum -= (atob * btoa * 2) * res[i];
            }
            BigInteger zeroatob = treeDP.Query(edges[eqIndex][0], edges[eqIndex][1]);
            BigInteger zerobtoa = treeDP.Query(edges[eqIndex][1], edges[eqIndex][0]);
            res[eqIndex] = (long)(sum / (zeroatob * zerobtoa * 2));
        }
        Console.WriteLine(string.Join("\n", res));
    }
}

class OmniTreeDP
{
    int[][] dp;
    bool[][] Calculated;
    List<int>[] Neighbours;
    Dictionary<int, int>[] IDToIndex;
    public OmniTreeDP(List<int>[] neighbours)
    {
        dp = new int[neighbours.Length][];
        Calculated = new bool[neighbours.Length][];
        IDToIndex = new Dictionary<int, int>[neighbours.Length];
        Neighbours = neighbours;
        for (int i = 0; i < neighbours.Length; i++)
        {
            dp[i] = new int[neighbours[i].Count];
            Calculated[i] = new bool[neighbours[i].Count];
            IDToIndex[i] = neighbours[i].Select((x, y) => new Tuple<int, int>(x, y)).ToDictionary(x => x.Item1, x => x.Item2);
        }
    }
    public int Query(int x, int edgeDist)
    {
        var index =  IDToIndex[edgeDist][x];
        if (Calculated[edgeDist][index]) return dp[edgeDist][index];
        Calculated[edgeDist][index] = true;
        return dp[edgeDist][index] = DFS(x, edgeDist);
    }
    public int DFS(int x, int from)
    {
        int res = 1/*value of node x*/;
        for (int i = 0; i < Neighbours[x].Count; i++)
        {
            if (Neighbours[x][i] == from) continue;
            int dfsRes = Calculated[x][i] ? dp[x][i] : dp[x][i] = DFS(Neighbours[x][i], x);
            Calculated[x][i] = true;
            res = res + dfsRes;
        }
        return res;
    }
}