103027: [Atcoder]ABC302 Ex - Ball Collector

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.

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.

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.

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

问题描述

我们有一个有$N$个顶点的树。第$(1 \le i \le N-1)$条边是一个连接顶点$U_i$和$V_i$的无向边。第$(1 \le i \le N)$个顶点有一个写有$A_i$的球和另一个写有$B_i$的球。

对每个$v = 2,3,\dots,N$，回答以下问题。（每个查询都是独立的。）

• 考虑从顶点$1$沿最短路径到达顶点$v$。每次访问一个顶点（包括顶点$1$和$v$），你都会捡起那里放置的一个球。找出写在捡起的球上的不同整数的最大数量。

限制条件

• $2 \le N \le 2 \times 10^5$
• $1 \le A_i,B_i \le N$
• 给定的图是一棵树。
• 输入中的所有值都是整数。

输入

输入以以下格式从标准输入给出：

$N$
$A_1$ $B_1$
$A_2$ $B_2$
$\vdots$
$A_N$ $B_N$
$U_1$ $V_1$
$U_2$ $V_2$
$\vdots$
$U_{N-1}$ $V_{N-1}$

输出

以空格分隔的方式打印出对于$v=2,3,\dots,N$的答案。

样例输入 1

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

样例输出 1

2 3 3

例如，当$v=4$时，您访问顶点$1,2,3$和$4$。通过选择写有$A_1,B_2,B_3,B_4(=1,3,1,2)$的球，球上的不同整数的数量为$3$，这是最大的。

样例输入 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

样例输出 2

4 3 2 3 4 3 4 2 3