A15410. Reset K Edges

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

You can perform the following operation at most $k$ times:

- choose an edge $(v, u)$ of the tree such that $v$ is a parent of $u$ ;
- remove the edge $(v, u)$ ;
- add an edge $(1, u)$ (i. e. make $u$ with its subtree a child of the root).

The height of a tree is the maximum depth of its vertices, and the depth of a vertex is the number of edges on the path from the root to it. For example, the depth of vertex $1$ is $0$ , since it's the root, and the depth of all its children is $1$ .

What's the smallest height of the tree that can be achieved?

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of testcases.

The first line of each testcase contains two integers $n$ and $k$ ( $2 \le n \le 2 \cdot 10^5$ ; $0 \le k \le n - 1$ ) — the number of vertices in the tree and the maximum number of operations you can perform.

The second line contains $n-1$ integers $p_2, p_3, \dots, p_n$ ( $1 \le p_i < i$ ) — the parent of the $i$ -th vertex. Vertex $1$ is the root.

The sum of $n$ over all testcases doesn't exceed $2 \cdot 10^5$ .

For each testcase, print a single integer — the smallest height of the tree that can achieved by performing at most $k$ operations.