CF1328E.Tree Queries

You are given a rooted tree consisting of $n$ vertices numbered from $1$ to $n$ . The root of the tree is a vertex number $1$ .

A tree is a connected undirected graph with $n-1$ edges.

You are given $m$ queries. The $i$ -th query consists of the set of $k_i$ distinct vertices $v_i[1], v_i[2], \dots, v_i[k_i]$ . Your task is to say if there is a path from the root to some vertex $u$ such that each of the given $k$ vertices is either belongs to this path or has the distance $1$ to some vertex of this path.

The first line of the input contains two integers $n$ and $m$ ( $2 \le n \le 2 \cdot 10^5$ , $1 \le m \le 2 \cdot 10^5$ ) — the number of vertices in the tree and the number of queries.

Each of the next $n-1$ lines describes an edge of the tree. Edge $i$ is denoted by two integers $u_i$ and $v_i$ , the labels of vertices it connects $(1 \le u_i, v_i \le n, u_i \ne v_i$ ).

It is guaranteed that the given edges form a tree.

The next $m$ lines describe queries. The $i$ -th line describes the $i$ -th query and starts with the integer $k_i$ ( $1 \le k_i \le n$ ) — the number of vertices in the current query. Then $k_i$ integers follow: $v_i[1], v_i[2], \dots, v_i[k_i]$ ( $1 \le v_i[j] \le n$ ), where $v_i[j]$ is the $j$ -th vertex of the $i$ -th query.

It is guaranteed that all vertices in a single query are distinct.

It is guaranteed that the sum of $k_i$ does not exceed $2 \cdot 10^5$ ( $\sum\limits_{i=1}^{m} k_i \le 2 \cdot 10^5$ ).

For each query, print the answer — "YES", if there is a path from the root to some vertex $u$ such that each of the given $k$ vertices is either belongs to this path or has the distance $1$ to some vertex of this path and "NO" otherwise.

The picture corresponding to the example:

Consider the queries.

The first query is $[3, 8, 9, 10]$ . The answer is "YES" as you can choose the path from the root $1$ to the vertex $u=10$ . Then vertices $[3, 9, 10]$ belong to the path from $1$ to $10$ and the vertex $8$ has distance $1$ to the vertex $7$ which also belongs to this path.

The second query is $[2, 4, 6]$ . The answer is "YES" as you can choose the path to the vertex $u=2$ . Then the vertex $4$ has distance $1$ to the vertex $1$ which belongs to this path and the vertex $6$ has distance $1$ to the vertex $2$ which belongs to this path.

The third query is $[2, 1, 5]$ . The answer is "YES" as you can choose the path to the vertex $u=5$ and all vertices of the query belong to this path.

The fourth query is $[4, 8, 2]$ . The answer is "YES" as you can choose the path to the vertex $u=9$ so vertices $2$ and $4$ both have distance $1$ to the vertex $1$ which belongs to this path and the vertex $8$ has distance $1$ to the vertex $7$ which belongs to this path.

The fifth and the sixth queries both have answer "NO" because you cannot choose suitable vertex $u$ .