<p>You are given the <code>root</code> of a binary tree with <strong>unique</strong> values, and an integer <code>start</code>. At minute <code>0</code>, an <strong>infection</strong> starts from the node with value <code>start</code>.</p>

<p>Each minute, a node becomes infected if:</p>

<ul>
	<li>The node is currently uninfected.</li>
	<li>The node is adjacent to an infected node.</li>
</ul>

<p>Return <em>the number of minutes needed for the entire tree to be infected.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/06/25/image-20220625231744-1.png" style="width: 400px; height: 306px;" />
<pre>
<strong>Input:</strong> root = [1,5,3,null,4,10,6,9,2], start = 3
<strong>Output:</strong> 4
<strong>Explanation:</strong> The following nodes are infected during:
- Minute 0: Node 3
- Minute 1: Nodes 1, 10 and 6
- Minute 2: Node 5
- Minute 3: Node 4
- Minute 4: Nodes 9 and 2
It takes 4 minutes for the whole tree to be infected so we return 4.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/06/25/image-20220625231812-2.png" style="width: 75px; height: 66px;" />
<pre>
<strong>Input:</strong> root = [1], start = 1
<strong>Output:</strong> 0
<strong>Explanation:</strong> At minute 0, the only node in the tree is infected so we return 0.
</pre>

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

<ul>
	<li>The number of nodes in the tree is in the range <code>[1, 10<sup>5</sup>]</code>.</li>
	<li><code>1 &lt;= Node.val &lt;= 10<sup>5</sup></code></li>
	<li>Each node has a <strong>unique</strong> value.</li>
	<li>A node with a value of <code>start</code> exists in the tree.</li>
</ul>

class Solution:
    def parent(self, root, start):
        maps = {}
        maps[root] = None
        q = [root]
        nodeg = None

        while q:
            node = q.pop(0)
            if node.val == start:
                nodeg = node
            if node.left:
                q.append(node.left)
                maps[node.left] = node
            if node.right:
                q.append(node.right)
                maps[node.right] = node

        return maps, nodeg

    def amountOfTime(self, root, start):
        if not root:
            return 0

        p, target = self.parent(root, start)
        dist = 0
        visit = set()
        q = [target]
        visit.add(target)

        while q:
            lev_len = len(q)
            for _ in range(lev_len):
                node = q.pop(0)
                if node.left and node.left not in visit:
                    q.append(node.left)
                    visit.add(node.left)
                if node.right and node.right not in visit:
                    q.append(node.right)
                    visit.add(node.right)
                if p[node] and p[node] not in visit:
                    q.append(p[node])
                    visit.add(p[node])
            dist += 1

        return dist - 1  # Adjust for the final increment

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