<p>There is a <strong>binary</strong> tree rooted at <code>0</code> consisting of <code>n</code> nodes. The nodes are labeled from <code>0</code> to <code>n - 1</code>. You are given a <strong>0-indexed</strong> integer array <code>parents</code> representing the tree, where <code>parents[i]</code> is the parent of node <code>i</code>. Since node <code>0</code> is the root, <code>parents[0] == -1</code>.</p>

<p>Each node has a <strong>score</strong>. To find the score of a node, consider if the node and the edges connected to it were <strong>removed</strong>. The tree would become one or more <strong>non-empty</strong> subtrees. The <strong>size</strong> of a subtree is the number of the nodes in it. The <strong>score</strong> of the node is the <strong>product of the sizes</strong> of all those subtrees.</p>

<p>Return <em>the <strong>number</strong> of nodes that have the <strong>highest score</strong></em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="example-1" src="https://assets.leetcode.com/uploads/2021/10/03/example-1.png" style="width: 604px; height: 266px;" />
<pre>
<strong>Input:</strong> parents = [-1,2,0,2,0]
<strong>Output:</strong> 3
<strong>Explanation:</strong>
- The score of node 0 is: 3 * 1 = 3
- The score of node 1 is: 4 = 4
- The score of node 2 is: 1 * 1 * 2 = 2
- The score of node 3 is: 4 = 4
- The score of node 4 is: 4 = 4
The highest score is 4, and three nodes (node 1, node 3, and node 4) have the highest score.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="example-2" src="https://assets.leetcode.com/uploads/2021/10/03/example-2.png" style="width: 95px; height: 143px;" />
<pre>
<strong>Input:</strong> parents = [-1,2,0]
<strong>Output:</strong> 2
<strong>Explanation:</strong>
- The score of node 0 is: 2 = 2
- The score of node 1 is: 2 = 2
- The score of node 2 is: 1 * 1 = 1
The highest score is 2, and two nodes (node 0 and node 1) have the highest score.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>n == parents.length</code></li>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>parents[0] == -1</code></li>
	<li><code>0 &lt;= parents[i] &lt;= n - 1</code> for <code>i != 0</code></li>
	<li><code>parents</code> represents a valid binary tree.</li>
</ul>

class Solution:
    def countHighestScoreNodes(self, parents: List[int]) -> int:
        from collections import defaultdict, Counter
        import math
        
        n = len(parents)
        graph = defaultdict(list)
        for i in range(1, n):
            graph[parents[i]].append(i)
        
        subtree_sizes = [0] * n
        
        def dfs(node):
            size = 1
            for child in graph[node]:
                size += dfs(child)
            subtree_sizes[node] = size
            return size
        
        dfs(0)
        
        max_score = 0
        score_counter = Counter()
        
        for node in range(n):
            score = 1
            for child in graph[node]:
                score *= subtree_sizes[child]
            if node != 0:
                score *= n - subtree_sizes[node]
            max_score = max(max_score, score)
            score_counter[score] += 1
        
        return score_counter[max_score]
60242