Khalin Graph

Little George Khalin designed a new type of graphs. He doesn't like the letter K, so he dropped it.

Preorder of a tree (sometimes called time-in order) is obtained using the following procedure:

Let's fix the root of the tree, and direct all edges away from the root. Preorder of the subtree of a vertex $v$ is $v$ followed by preorders of subtrees of all its children (if any) in some order. A preorder of a tree with a fixed root is any preorder of the subtree of the root.

Note that there are multiple preorders of the same tree, since a preorder depends on the choice of the root, as well as the order in which children subtrees are considered at every vertex.

A Halin graph is a graph obtained using the following procedure: \\ There is a tree that we will call the base tree of the graph, which has at least 4 vertices and has no vertices of degree 2. One of its preorders is specified. The root of the tree with respect to this preorder is not a leaf.

Let $v\_1, v\_2, ..., v\_m$ be the leaves of the tree in order they appear in the preorder. For each $i$ from $1$ to $m$, add an edge between the vertices $v\_i$ and $v\_{(i \bmod m) + 1}$ to the tree. Those edges are called additional. The resulting graph is the Halin graph with respect to the base tree and the specified preorder.

A 3-matching of a graph $G$ is a set of edges $S$ such that the connected components of the graph formed by removal of all edges not in $S$ from $G$ are trees of size 3 or 1.

You are given a Halin graph. Find the number of its 3-matchings modulo $998\\,244\\,353.$

The first line contains a single integer $n$ ($4 \leq n \leq 10^5$) the number of vertices in the base tree. Vertices are enumerated according to the preorder.

The second line contains $n-1$ integers. $i$-th of them is $p\_i$ ($1 \leq p\_i \leq i$), describing an edge between $p\_i$-th and $i+1$-th vertices in the base tree.

It is guaranteed that the base tree is a tree, has no vertices of degree 2, and that the vertex $1$ is not a leaf.

Output a single integer --- the number of 3-matchings of the given graph modulo $998\\,244\\,353$.

In the first example the actual Halin graph is the complete graph on four vertices.

In the second example the leaves are $\[2, 4, 5, 6]$, thus there are four additional edges --- (2, 4), (4, 5), (5, 6), (6, 2).

Note that there is no letter K in the meaningful parts of the statement.