Tree

Tree $T$ is made of vertices and edges. Each circle in Figure 1 is a vertex, and each line joining two vertices is an edge. The vertex drawn at the top is the root, and there is exactly one root. The $N$ vertices are numbered 1 through $N$, and the root is always vertex 1.

A path joining two vertices $v$ and $w$ is a sequence of vertices $(v_0, v_1, \dots, v_m)$ in which neighboring vertices $v_i$ and $v_{i+1}$ are joined by an edge, $v_0 = v$, and $v_m = w$. In a tree, exactly one such path exists between any two vertices. In Figure 1 the only path joining vertex 3 and vertex 11 is $(3, 4, 1, 7, 11)$.

Figure 1

Let $P$ be the unique path from vertex $v$ to the root. The vertex that is joined to $v$ by an edge and lies on $P$ is the parent of $v$. In Figure 1 the parent of 4, 7 and 9 is 1, and the parent of 2 and 11 is 7. The root has no parent.

When one edge is removed, pairs of vertices other than the two endpoints of that edge can also lose the path between them. Removing the edge between 7 and 11 in Figure 1 leaves no path between 8 and 5.

You are given the tree, and then queries and edge removals mixed in an arbitrary order. Carry out the operations in the given order and print the answer to every query.

The first line contains the number of vertices $N$ and the number of queries $Q$ ($1 \le N, Q \le 200\,000$).

The $i$-th of the next $N-1$ lines contains one integer $a$, the parent of vertex $i+1$ ($1 \le a \le N$). The given parent relation always forms a tree rooted at vertex 1.

Each of the next $Q$ lines contains three integers $b$, $c$ and $d$ ($1 \le b, c \le N$, and $d$ is 0 or 1). If $d = 0$, only handle the query asking whether a path joins $b$ and $c$. If $d = 1$, handle the same query first, then remove the edge joining $b$ and the parent of $b$ when the answer is YES, or the edge joining $c$ and the parent of $c$ when the answer is NO. If the edge to remove does not exist or was already removed, no edge is removed.

Print the answers to the $Q$ queries in order, one per line. Print YES when a path exists and NO when it does not.