<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>

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
import collections
class Solution:
    def amountOfTime(self, root: Optional[TreeNode], start: int) -> int:
        hm = collections.defaultdict(list)

        def bfstomap(root, start, hm):
            qu = []
            qu.append(root)
            st = None
            # BFS to map parent nodes and find the start node
            while len(qu) != 0:
                node = qu.pop(0)
                if node.val == start:
                    st = node
                if node.left is not None:
                    hm[node.left] = node  # Map the parent
                    qu.append(node.left)  # Add left child to queue
                if node.right is not None:
                    hm[node.right] = node  # Map the parent
                    qu.append(node.right)  # Add right child to queue
            return st

        def mintb(sta, hm):
            vis = collections.defaultdict(bool)  # Boolean visited map
            qu = []
            qu.append(sta)
            vis[sta] = True  # Mark the start node as visited
            t = 0
            while len(qu) != 0:
                s = len(qu)
                fl = False
                for i in range(s):
                    curr = qu.pop(0)
                    # Check left child
                    if curr.left is not None and not vis[curr.left]:
                        qu.append(curr.left)
                        vis[curr.left] = True
                        fl = True
                    # Check right child
                    if curr.right is not None and not vis[curr.right]:
                        qu.append(curr.right)
                        vis[curr.right] = True
                        fl = True
                    # Check parent node
                    if curr in hm and not vis[hm[curr]]:
                        qu.append(hm[curr])
                        vis[hm[curr]] = True
                        fl = True
                if fl:
                    t += 1  # Increment time if any node is infected
            return t

        sta = bfstomap(root, start, hm)  # Find start node and build parent map
        time = mintb(sta, hm)  # Calculate time to infect all nodes
        return time
        
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