Tree Queries
Description
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.
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.
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 $x \cdot q \equiv p \pmod{M}$.
Input
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.
Output
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 $x \cdot q \equiv p \pmod{M}$.
Samples
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
1
199648871
399297742
199648871
199648871
598946614
199648873
2
2
2
Note
The image below shows the tree in the example:
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 ($2.2, 2.4, 2, 2, 2$).