[ABC264Ex] Perfect Binary Tree

Problem Statement

We have a rooted tree with $N$ vertices numbered $1,2,\dots,N$.

The tree is rooted at Vertex $1$, and the parent of Vertex $i \ge 2$ is Vertex $P_i(<i)$.

For each integer $k=1,2,\dots,N$, solve the following problem:

There are $2^{k-1}$ ways to choose some of the vertices numbered between $1$ and $k$ so that Vertex $1$ is chosen.

How many of them satisfy the following condition: the subgraph induced by the set of chosen vertices forms a perfect binary tree (with $2^d-1$ vertices for a positive integer $d$) rooted at Vertex $1$?

Since the count may be enormous, print the count modulo $998244353$.

What is an induced subgraph?

Let $S$ be a subset of the vertex set of a graph $G$. The subgraph $H$ induced by this vertex set $S$ is constructed as follows:

• Let the vertex set of $H$ equal $S$.
• Then, we add edges to $H$ as follows:
• For all vertex pairs $(i, j)$ such that $i,j \in S, i < j$, if there is an edge connecting $i$ and $j$ in $G$, then add an edge connecting $i$ and $j$ to $H$.

What is a perfect binary tree?

A perfect binary tree is a rooted tree that satisfies all of the following conditions:

• Every vertex that is not a leaf has exactly $2$ children.
• All leaves have the same distance from the root.

Here, we regard a graph with \(1\) vertex and \(0\) edges as a perfect binary tree, too.

Constraints

• All values in input are integers.
• $1 \le N \le 3 \times 10^5$
• $1 \le P_i < i$

---

Input

Input is given from Standard Input in the following format:

$N$
$P_2$ $P_3$ $\dots$ $P_N$

Output

Print $N$ lines.
The $i$-th ($1 \le i \le N$) line should contain the answer as an integer when $k=i$.

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Sample Input 1

10
1 1 2 1 2 5 5 5 1

Sample Output 1

1
1
2
2
4
4
4
5
7
10

The following ways of choosing vertices should be counted:

• $\{1\}$ when $k \ge 1$
• $\{1,2,3\}$ when $k \ge 3$
• $\{1,2,5\},\{1,3,5\}$ when $k \ge 5$
• $\{1,2,4,5,6,7,8\}$ when $k \ge 8$
• $\{1,2,4,5,6,7,9\},\{1,2,4,5,6,8,9\}$ when $k \ge 9$
• $\{1,2,10\},\{1,3,10\},\{1,5,10\}$ when $k = 10$

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Sample Input 2

1

Sample Output 2

1

If $N=1$, the $2$-nd line of the Input is empty.

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Sample Input 3

10
1 2 3 4 5 6 7 8 9

Sample Output 3

1
1
1
1
1
1
1
1
1
1

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Sample Input 4

13
1 1 1 2 2 2 3 3 3 4 4 4

Sample Output 4

1
1
2
4
4
4
4
4
7
13
13
19
31

先考虑如果不带实时询问怎么做？定义 \(dp_{i,j}\) 为以 \(i\) 为节点，深度为 \(j\) 的导出完全二叉树有多少个。发现 \(j\) 是 \(logn\) 级别的，因为一个 \(j\) 层完全二叉树的节点数量是 \(2^j\) 级别，而节点数量要 \(\le n\)。

用 \(son\) 表示 \(i\) 的所有儿子的集合， 初始化 \(dp_{i,0}=1\)$$dp_{i,j}=\sum\limits_{v1\in son}\sum\limits_{v2>v1,v2\in son}dp_{v1,j-1}\times dp_{v2,j-1}$$

这个可以化简为 $$(\sum\limits_{v\in son}dp_{v,j-1})^2-\sum\limits_{v\in son}dp_{v,j-1}^2$$

这个东西就可以实现树上 \(O(1)\) 转移了，最终答案为 \(\sum\limits_{j=0}^{logn}dp_{1,j}\)，总复杂度 \(O(nlogn)\)

现在要求加点，询问。那么思路很简单，首先如果一个点的层数大过 \(logn\)，他影响不到答案。然后考虑不断往上爬，去更改会改变的答案。

要直接维护 \(dp\) 值不容易，考虑维护所有 \(dp\) 值的和还有平方和，有这两个更改我们可以推出 \(dp\) 值的更改。然后发现，如果现在加入点 \(x\) 的时候，点 \(y\) 与点 \(x\) 的层数差为 \(a\)，那么只有 \(dp_{y,a}\) 有可能更改，加上 \(x\) 的层数 \(\le logn\)，所以这样子改的复杂度是 \(O(logn)\) 的。具体更改时就是减去旧的加上新的就行了。

#include<cstdio>
const int N=3e5+5,P=998244353,inv2=499122177;
typedef long long LL;
long long s[N][25],f[N][25],dp[N][25],ans,dep[N];//s表示和，f表示平方和
int n,k,fa[N];
void dfs(int x,int y,LL a,LL b)//a表示原来的，b表示新的
{
	LL k=dp[x][y];
	(s[x][y]+=b-a+P)%=P;
	(f[x][y]+=b*b%P-a*a%P+P)%=P;
	dp[x][y]=(s[x][y]*s[x][y]%P-f[x][y]+P)*inv2%P;
	if(x-1)
		dfs(fa[x],y+1,k,dp[x][y]);
}
int main()
{
	scanf("%d",&n);
	dp[1][0]=1;
	puts("1");
	for(int i=2;i<=n;i++)
	{
		scanf("%d",fa+i),dep[i]=dep[fa[i]]+1;
		f[i][0]=dp[i][0]=s[i][0]=1;
		if(dep[i]<=20)
			dfs(fa[i],1,0,1);
//		puts("qzmakioi");
		ans=0;
		for(int j=0;j<=20;j++)
			(ans+=dp[1][j])%=P;
		printf("%lld\n",ans);
	}
}