Score : $625$ points

问题描述

我们拥有一棵包含 $N$ 个顶点的树。第 $i$-条边（其中 $1 \le i \le N-1$）是从顶点 $U_i$ 到顶点 $V_i$ 的一条无向边。第 $i$ 个顶点（其中 $1 \le i \le N$）上有一个写有数字 $A_i$ 的球和另一个写有数字 $B_i$ 的球。

对于每个顶点 $v = 2,3,\dots,N$，独立回答以下问题：

• 考虑沿最短路径从顶点 $1$ 行走到顶点 $v$。每次访问一个顶点（包括顶点 $1$ 和 $v$），你都会拾取该处放置的一个球。找出所拾取球上所写整数的最大不同数值数量。

以上为通义千问 qwen-max 翻译，仅供参考。

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

update @ 2024/3/10 08:30:28