CF1665C.Tree Infection

A tree is a connected graph without cycles. A rooted tree has a special vertex called the root. The parent of a vertex $v$ (different from root) is the previous to $v$ vertex on the shortest path from the root to the vertex $v$. Children of the vertex $v$ are all vertices for which $v$ is the parent.

You are given a rooted tree with $n$ vertices. The vertex $1$ is the root. Initially, all vertices are healthy.

Each second you do two operations, the spreading operation and, after that, the injection operation:

1. Spreading: for each vertex $v$, if at least one child of $v$ is infected, you can spread the disease by infecting at most one other child of $v$ of your choice.
2. Injection: you can choose any healthy vertex and infect it.

This process repeats each second until the whole tree is infected. You need to find the minimal number of seconds needed to infect the whole tree.

树是无环的连通图。有根树具有一个特殊的顶点，称为根。顶点 $v$（$v$ 不为根）的父节点是指从根到 $v$ 的最短路径上位于 $v$ 之前的那个顶点。顶点 $v$ 的子节点是指所有以 $v$ 为父节点的顶点。

给你一棵含 $n$ 个顶点的有根树，其中顶点 $1$ 是根。初始时，所有顶点均为健康状态。

每秒钟你执行两个操作：传播操作，随后是注入操作：

1. 传播操作：对每个顶点 $v$，若 $v$ 至少有一个子节点已被感染，则你可以选择至多一个 $v$ 的其他（尚未感染的）子节点将其感染；
2. 注入操作：你可以任选一个健康顶点并将其感染。

该过程每秒重复一次，直至整棵树全部被感染。你需要求出感染整棵树所需的最少秒数。

The input consists of multiple test cases. The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the number of the vertices in the given tree.

The second line of each test case contains $n - 1$ integers $p_2, p_3, \ldots, p_n$ ($1 \le p_i \le n$), where $p_i$ is the ancestor of the $i$-th vertex in the tree.

It is guaranteed that the given graph is a tree.

It is guaranteed that the sum of $n$ over all test cases doesn't exceed $2 \cdot 10^5$.

输入包含多个测试用例。第一行包含一个整数 $t$（$1 \le t \le 10^4$），表示测试用例的数量。随后是各测试用例的描述。

每个测试用例的第一行包含一个整数 $n$（$2 \le n \le 2 \cdot 10^5$），表示给定树中顶点的数量。

每个测试用例的第二行包含 $n - 1$ 个整数 $p_2, p_3, \ldots, p_n$（$1 \le p_i \le n$），其中 $p_i$ 表示树中第 $i$ 个顶点的父节点（祖先）。

保证所给图是一棵树。

保证所有测试用例的 $n$ 之和不超过 $2 \cdot 10^5$。

For each test case you should output a single integer — the minimal number of seconds needed to infect the whole tree.

对于每个测试用例，你应该输出一个整数——感染整棵树所需的最少秒数。

The image depicts the tree from the first test case during each second.

A vertex is black if it is not infected. A vertex is blue if it is infected by injection during the previous second. A vertex is green if it is infected by spreading during the previous second. A vertex is red if it is infected earlier than the previous second.

Note that you are able to choose which vertices are infected by spreading and by injections.

该图像展示了第一个测试用例中的树在每一秒时的状态。

• 若一个顶点未被感染，则其为黑色；
• 若一个顶点在前一秒通过注射方式被感染，则其为蓝色；
• 若一个顶点在前一秒通过传播方式被感染，则其为绿色；
• 若一个顶点在前一秒之前即已被感染，则其为红色。

注意：你可以自主选择哪些顶点通过传播方式被感染、哪些顶点通过注射方式被感染。