CF1056D.Decorate Apple Tree

There is one apple tree in Arkady's garden. It can be represented as a set of junctions connected with branches so that there is only one way to reach any junctions from any other one using branches. The junctions are enumerated from $1$ to $n$, the junction $1$ is called the root.

A subtree of a junction $v$ is a set of junctions $u$ such that the path from $u$ to the root must pass through $v$. Note that $v$ itself is included in a subtree of $v$.

A leaf is such a junction that its subtree contains exactly one junction.

The New Year is coming, so Arkady wants to decorate the tree. He will put a light bulb of some color on each leaf junction and then count the number happy junctions. A happy junction is such a junction $t$ that all light bulbs in the subtree of $t$ have different colors.

Arkady is interested in the following question: for each $k$ from $1$ to $n$, what is the minimum number of different colors needed to make the number of happy junctions be greater than or equal to $k$?

阿卡迪的花园里有一棵苹果树。这棵树可以表示为一组由树枝连接的节点，使得任意两个节点之间都恰好存在一条路径。节点编号从 $1$ 到 $n$，其中节点 $1$ 被称为根节点。

节点 $v$ 的子树是指所有满足“从 $u$ 到根节点的路径必经过 $v$”的节点 $u$ 构成的集合。注意：$v$ 自身也属于其自身的子树。

叶子节点是指其子树中恰好包含一个节点的节点。

新年将至，阿卡迪打算装饰这棵树。他将在每个叶子节点上放置一个某种颜色的灯泡，然后统计“快乐节点”的数量。一个节点 $t$ 被称为快乐节点，当且仅当其子树内所有灯泡的颜色互不相同。

阿卡迪关心如下问题：对每个 $k = 1, 2, \dots, n$，要使快乐节点的数量不少于 $k$，所需的最少颜色种类数是多少？

The first line contains a single integer $n$ ($1 \le n \le 10^5$) — the number of junctions in the tree.

The second line contains $n - 1$ integers $p_2$, $p_3$, ..., $p_n$ ($1 \le p_i \lt i$), where $p_i$ means there is a branch between junctions $i$ and $p_i$. It is guaranteed that this set of branches forms a tree.

第一行包含一个整数 $n$（$1 \le n \le 10^5$）—— 表示树中节点的数量。

第二行包含 $n - 1$ 个整数 $p_2$, $p_3$, ..., $p_n$（$1 \le p_i \lt i$），其中 $p_i$ 表示节点 $i$ 与节点 $p_i$ 之间存在一条边。保证这些边构成一棵树。

Output $n$ integers. The $i$-th of them should be the minimum number of colors needed to make the number of happy junctions be at least $i$.

输出 $n$ 个整数。其中第 $i$ 个整数应为使得快乐交叉点数量至少为 $i$ 所需的最少颜色数。

In the first example for $k = 1$ and $k = 2$ we can use only one color: the junctions $2$ and $3$ will be happy. For $k = 3$ you have to put the bulbs of different colors to make all the junctions happy.

In the second example for $k = 4$ you can, for example, put the bulbs of color $1$ in junctions $2$ and $4$, and a bulb of color $2$ into junction $5$. The happy junctions are the ones with indices $2$, $3$, $4$ and $5$ then.

在第一个例子中，当 $k = 1$ 和 $k = 2$ 时，我们只能使用一种颜色：结点 $2$ 和 $3$ 将是快乐的。当 $k = 3$ 时，必须在结点上安装不同颜色的灯泡，才能使所有结点都快乐。

在第二个例子中，当 $k = 4$ 时，例如可将颜色为 $1$ 的灯泡安装在结点 $2$ 和 $4$ 上，并将颜色为 $2$ 的灯泡安装在结点 $5$ 上。此时，快乐的结点为索引为 $2$、$3$、$4$ 和 $5$ 的结点。