CF1481F.AB Tree

Kilani and Abd are neighbors for 3000 years, but then the day came and Kilani decided to move to another house. As a farewell gift, Kilani is going to challenge Abd with a problem written by their other neighbor with the same name Abd.

The problem is:

You are given a connected tree rooted at node $1$.

You should assign a character a or b to every node in the tree so that the total number of a's is equal to $x$ and the total number of b's is equal to $n - x$.

Let's define a string for each node $v$ of the tree as follows:

• if $v$ is root then the string is just one character assigned to $v$:
• otherwise, let's take a string defined for the $v$'s parent $p_v$ and add to the end of it a character assigned to $v$.

You should assign every node a character in a way that minimizes the number of distinct strings among the strings of all nodes.

基拉尼和阿卜杜是长达3000年的邻居，但这一天终于到来，基拉尼决定搬去另一所房子。作为临别礼物，基拉尼将用他们另一位同名邻居阿卜杜所出的一道题来挑战阿卜杜。

题目如下：

给定一棵以节点 $1$ 为根的连通树。

你需要为树中每个节点分配字符 a 或 b，使得 a 的总个数恰好为 $x$，b 的总个数恰好为 $n - x$。

对树中每个节点 $v$，我们定义一个字符串如下：

• 若 $v$ 是根节点，则该字符串即为分配给 $v$ 的单个字符；
• 否则，取 $v$ 的父节点 $p_v$ 所对应的字符串，并在其末尾添加分配给 $v$ 的字符。

你需要为每个节点分配字符，使得所有节点对应字符串中互不相同的字符串总数最小。

The first line contains two integers $n$ and $x$ ($1 \leq n \leq 10^5$; $0 \leq x \leq n$) — the number of vertices in the tree the number of a's.

The second line contains $n - 1$ integers $p_2, p_3, \dots, p_{n}$ ($1 \leq p_i \leq n$; $p_i \neq i$), where $p_i$ is the parent of node $i$.

It is guaranteed that the input describes a connected tree.

第一行包含两个整数 $n$ 和 $x$（$1 \leq n \leq 10^5$；$0 \leq x \leq n$）—— 分别表示树中顶点的数量和字母 a 的数量。

第二行包含 $n - 1$ 个整数 $p_2, p_3, \dots, p_{n}$（$1 \leq p_i \leq n$；$p_i \neq i$），其中 $p_i$ 表示节点 $i$ 的父节点。

保证输入描述的是一棵连通的树。

In the first line, print the minimum possible total number of distinct strings.

In the second line, print $n$ characters, where all characters are either a or b and the $i$-th character is the character assigned to the $i$-th node.

Make sure that the total number of a's is equal to $x$ and the total number of b's is equal to $n - x$.

If there is more than one answer you can print any of them.

第一行输出可能的最少不同字符串总数。

第二行输出 $n$ 个字符，每个字符均为 a 或 b，其中第 $i$ 个字符表示分配给第 $i$ 个节点的字符。

请确保字符 a 的总数为 $x$，字符 b 的总数为 $n - x$。

若存在多个可行答案，输出任意一个即可。

The tree from the sample is shown below:

The tree after assigning characters to every node (according to the output) is the following:

Strings for all nodes are the following:

• string of node $1$ is: a
• string of node $2$ is: aa
• string of node $3$ is: aab
• string of node $4$ is: aab
• string of node $5$ is: aabb
• string of node $6$ is: aabb
• string of node $7$ is: aabb
• string of node $8$ is: aabb
• string of node $9$ is: aa

The set of unique strings is ${\text{a}, \text{aa}, \text{aab}, \text{aabb}}$, so the number of distinct strings is $4$.

样例中的树如下所示：

按照输出结果为每个节点分配字符后得到的树如下所示：

各节点对应的字符串如下：

• 节点 $1$ 的字符串为：a
• 节点 $2$ 的字符串为：aa
• 节点 $3$ 的字符串为：aab
• 节点 $4$ 的字符串为：aab
• 节点 $5$ 的字符串为：aabb
• 节点 $6$ 的字符串为：aabb
• 节点 $7$ 的字符串为：aabb
• 节点 $8$ 的字符串为：aabb
• 节点 $9$ 的字符串为：aa

所有不同字符串构成的集合为 ${\text{a}, \text{aa}, \text{aab}, \text{aabb}}$，因此不同字符串的个数为 $4$。