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 with the following procedure:

Fix the root of the tree and direct all edges away from the root. The preorder of the subtree of a vertex $v$ is $v$ followed by the preorders of the 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 a tree has multiple preorders, since a preorder depends on the choice of the root and on the order in which the children subtrees are considered at every vertex.

A Halin graph is obtained with the following procedure. There is a tree, which we call the base tree of the graph. It has at least 4 vertices and 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 $v_1, v_2, \ldots, v_m$ be the leaves of the tree in the order they appear in the preorder. For each $i$ from $1$ to $m$, add an edge between $v_i$ and $v_{(i \bmod m) + 1}$ to the tree. These 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 removing 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 numbered according to the preorder.

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

It is guaranteed that the base tree is a tree, has no vertices of degree 2, and 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 $[2, 4, 5, 6]$, so 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.