407674: GYM102870 D Data Structure Master and Orz Pandas

Master wang9897 is the First Data Structure Master in Xidian University. Today he is teaching some Orz Pandas to learn Link-Cut Tree (LCT). As a practice, wang9897 has given an interesting problem to the Orz Pandas.

Note that you don't need to know LCT to solve this problem.

Given a rooted tree with $$$n$$$ nodes whose root is numbered $$$1$$$. In LCT each node $$$u$$$ can be assigned one preferred child $$$v$$$. Certainly $$$v$$$ must be a child of $$$u$$$. Initially there is no preferred child assigned to each node.

Then $$$m$$$ "LCT access" operations are performed as follow. For each operation, a node $$$v$$$ is chosen from all nodes uniformly. The path from the root to $$$v$$$, $$$(v_1,v_2,\cdots,v_k)$$$ is then found, where $$$v_1 = 1$$$, and $$$v_k = v$$$. For $$$i = 1, 2, \cdots, k - 1$$$, if $$$v_i$$$ is not assigned a preferred child, or it is assigned to a preferred child which is not $$$v_{i+1}$$$, the preferred child of $$$v_i$$$ is reassigned to $$$v_{i+1}$$$.

Let $$$f(m)$$$ be the time of reassignments of preferred child in total. Master wang9897 want the Orz Pandas to calculate $$$$$$ \lim_{m\rightarrow\infty} \frac{f(m)}{m} $$$$$$

Please solve the problem so wang9897 can compare the Orz Pandas' answer to yours.

There is only one test case in the input file.

The first line contains a integer $$$n$$$, the size of the tree. The second line contains $$$n-1$$$ integers $$$p_2,p_3,\cdots,p_{n-1}$$$. $$$p_i$$$ is the parent of the node $$$i$$$.

It's guaranteed that $$$1 \leq p_i \leq n \leq 10^5$$$.

Output one line, containing an integer, the answer modulo $$$998244353$$$.

5
1 1 2 2

499122177

For the sample, the answer is $$$1 / 2$$$, so $$$1 \cdot 2^{-1}$$$ should be outputed.