<p>You are given a <strong>tree</strong> (i.e. a connected, undirected graph that has no cycles) <strong>rooted</strong> at node <code>0</code> consisting of <code>n</code> nodes numbered from <code>0</code> to <code>n - 1</code>. The tree is represented by a <strong>0-indexed</strong> array <code>parent</code> of size <code>n</code>, where <code>parent[i]</code> is the parent of node <code>i</code>. Since node <code>0</code> is the root, <code>parent[0] == -1</code>.</p>

<p>You are also given a string <code>s</code> of length <code>n</code>, where <code>s[i]</code> is the character assigned to node <code>i</code>.</p>

<p>Return <em>the length of the <strong>longest path</strong> in the tree such that no pair of <strong>adjacent</strong> nodes on the path have the same character assigned to them.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/03/25/testingdrawio.png" style="width: 201px; height: 241px;" />
<pre>
<strong>Input:</strong> parent = [-1,0,0,1,1,2], s = &quot;abacbe&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> The longest path where each two adjacent nodes have different characters in the tree is the path: 0 -&gt; 1 -&gt; 3. The length of this path is 3, so 3 is returned.
It can be proven that there is no longer path that satisfies the conditions. 
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/03/25/graph2drawio.png" style="width: 201px; height: 221px;" />
<pre>
<strong>Input:</strong> parent = [-1,0,0,0], s = &quot;aabc&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> The longest path where each two adjacent nodes have different characters is the path: 2 -&gt; 0 -&gt; 3. The length of this path is 3, so 3 is returned.
</pre>

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

<ul>
	<li><code>n == parent.length == s.length</code></li>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= parent[i] &lt;= n - 1</code> for all <code>i &gt;= 1</code></li>
	<li><code>parent[0] == -1</code></li>
	<li><code>parent</code> represents a valid tree.</li>
	<li><code>s</code> consists of only lowercase English letters.</li>
</ul>

class Solution:
    # topo sort BFS, start from child nodes, remember the longest 2 path lengths for each node
    def longestPath(self, A: List[int], s: str) -> int:
        n = len(A)
        child_num = [0] * n  # child degree 
        for a in A[1:]:
            child_num[a] += 1
        
        # The longest valid chain with node i as parent.
        longest = [[0] for _ in range(n)]
        
        # Add all leaf nodes to deque, [current node, the maximum length rooted at this node]
        dq = collections.deque()
        for i in range(n):
            if child_num[i] == 0:
                dq.append([i, 1])
        
        res = 1
        while dq:
            node, path_len = dq.popleft()
            parent = A[node] 
            # Minus the number of unvisited child node by 1
            child_num[parent] -= 1  
            if s[parent] != s[node]: 
                # insert & maintaining sorted order of longest[parent]
                bisect.insort_right(longest[parent], path_len) 
                if len(longest[parent]) > 2: # only maintain the top 2
                    longest[parent].pop(0)
                     
            if child_num[parent] == 0:
                res = max(res, 1 + sum(longest[parent]))
                # res = max(res, 1 + sum(longest[parent][-2:]))
                dq.append([parent, 1 + longest[parent][-1]])
                
        return res
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