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.

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.

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.

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 $4$ .