Tree Queries
题意翻译
给定一棵$N$个节点的树,有$Q$次操作
$1\space v\space d$ 给定一个点$v$和一个权值$d$,等概率地选择一个点$r$,对每一个点$u$,若$v$在$u$到$r$的路径上,则$u$的权值加上$d$ (权值一开始为$0$)
$2\space v$ 查询$v$的权值期望,对$998244353$取模
$1\leqslant N,Q \leqslant 150000$
题目描述
Hanh is a famous biologist. He loves growing trees and doing experiments on his own garden.
One day, he got a tree consisting of $ n $ vertices. Vertices are numbered from $ 1 $ to $ n $ . A tree with $ n $ vertices is an undirected connected graph with $ n-1 $ edges. Initially, Hanh sets the value of every vertex to $ 0 $ .
Now, Hanh performs $ q $ operations, each is either of the following types:
- Type $ 1 $ : Hanh selects a vertex $ v $ and an integer $ d $ . Then he chooses some vertex $ r $ uniformly at random, lists all vertices $ u $ such that the path from $ r $ to $ u $ passes through $ v $ . Hanh then increases the value of all such vertices $ u $ by $ d $ .
- Type $ 2 $ : Hanh selects a vertex $ v $ and calculates the expected value of $ v $ .
Since Hanh is good at biology but not math, he needs your help on these operations.
输入输出格式
输入格式
The first line contains two integers $ n $ and $ q $ ( $ 1 \leq n, q \leq 150\,000 $ ) — the number of vertices on Hanh's tree and the number of operations he performs.
Each of the next $ n - 1 $ lines contains two integers $ u $ and $ v $ ( $ 1 \leq u, v \leq n $ ), denoting that there is an edge connecting two vertices $ u $ and $ v $ . It is guaranteed that these $ n - 1 $ edges form a tree.
Each of the last $ q $ lines describes an operation in either formats:
- $ 1 $ $ v $ $ d $ ( $ 1 \leq v \leq n, 0 \leq d \leq 10^7 $ ), representing a first-type operation.
- $ 2 $ $ v $ ( $ 1 \leq v \leq n $ ), representing a second-type operation.
It is guaranteed that there is at least one query of the second type.
输出格式
For each operation of the second type, write the expected value on a single line.
Let $ M = 998244353 $ , it can be shown that the expected value can be expressed as an irreducible fraction $ \frac{p}{q} $ , where $ p $ and $ q $ are integers and $ q \not \equiv 0 \pmod{M} $ . Output the integer equal to $ p \cdot q^{-1} \bmod M $ . In other words, output such an integer $ x $ that $ 0 \le x < M $ and $ x \cdot q \equiv p \pmod{M} $ .
输入输出样例
输入样例 #1
5 12
1 2
1 3
2 4
2 5
1 1 1
2 1
2 2
2 3
2 4
2 5
1 2 2
2 1
2 2
2 3
2 4
2 5
输出样例 #1
1
199648871
399297742
199648871
199648871
598946614
199648873
2
2
2
说明
The image below shows the tree in the example:
![](https://cdn.luogu.com.cn/upload/vjudge_pic/CF1254D/8305f689c4b5cb634af4d0b9b9cf646ab39a9a7b.png)For the first query, where $ v = 1 $ and $ d = 1 $ :
- If $ r = 1 $ , the values of all vertices get increased.
- If $ r = 2 $ , the values of vertices $ 1 $ and $ 3 $ get increased.
- If $ r = 3 $ , the values of vertices $ 1 $ , $ 2 $ , $ 4 $ and $ 5 $ get increased.
- If $ r = 4 $ , the values of vertices $ 1 $ and $ 3 $ get increased.
- If $ r = 5 $ , the values of vertices $ 1 $ and $ 3 $ get increased.
Hence, the expected values of all vertices after this query are ( $ 1, 0.4, 0.8, 0.4, 0.4 $ ).
For the second query, where $ v = 2 $ and $ d = 2 $ :
- If $ r = 1 $ , the values of vertices $ 2 $ , $ 4 $ and $ 5 $ get increased.
- If $ r = 2 $ , the values of all vertices get increased.
- If $ r = 3 $ , the values of vertices $ 2 $ , $ 4 $ and $ 5 $ get increased.
- If $ r = 4 $ , the values of vertices $ 1 $ , $ 2 $ , $ 3 $ and $ 5 $ get increased.
- If $ r = 5 $ , the values of vertices $ 1 $ , $ 2 $ , $ 3 $ and $ 4 $ get increased.
Hence, the expected values of all vertices after this query are ( $ 2.2, 2.4, 2, 2, 2 $ ).