D. Reset K Edges
time limit per test
4 seconds
memory limit per test
256 megabytes
standard input
standard output

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.

5 1
1 1 2 2
5 2
1 1 2 2
6 0
1 2 3 4 5
6 1
1 2 3 4 5
4 3
1 1 1