Problem Statement

We have a tree with N vertices. The $i-th (1 \le i \le N-1)$ edge is an undirected edge between vertices U_i and $V_i. Vertex i (1 \le i \le N) has a$ ball with A_i written on it and another with B_i.
For each $v = 2,3,\dots,N,$ answer the following question. (Each query is independent.)

• Consider traveling from vertex 1 to vertex v on the shortest path. Every time you visit a vertex (including vertices 1 and v), you pick up one ball placed there. Find the maximum number of distinct integers written on the picked-up balls.

Input

The input is given from Standard Input in the following format:
N
A_1 B_1
A_2 B_2
$\vdots$
A_N B_N
U_1 V_1
U_2 V_2
$\vdots$
U_{N-1} V_{N-1}

Output

Print the answers for $v=2,3,\dots,N,$ separated by spaces.

Constraints

• $2 \le N \le 2 \times 10^5$
• $1 \le A_i,B_i \le N$
• The given graph is a tree.
• All values in the input are integers.

Sample Input 1

4
1 2
2 3
3 1
1 2
1 2
2 3
3 4

Sample Output 1

2 3 3

For example, when v=4, you visit vertices 1,2,3, and 4. By choosing balls with A_1,B_2,B_3,B_4(=1,3,1,2) written on them, the number of distinct integers on the balls is 3, which is the maximum.

Sample Input 2

10
2 5
2 2
8 8
4 3
6 10
8 1
9 10
1 7
9 3
5 10
9 3
1 9
3 6
4 1
3 8
10 9
5 4
7 2
9 7

Sample Output 2

4 3 2 3 4 3 4 2 3

见题面 Input 部分

见题面 Output 部分