<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
class BI_TreeNode:
    def __init__(self, val=0, left=None, right=None, parent=None):
        self.val = val
        self.left = left
        self.right = right
        self.parent = parent


class Solution:
    start_node = None

    def amountOfTime(self, root: Optional[TreeNode], start: int) -> int:
        new_root = self.convert_to_BI_tree_and_find_start_node(None, root, start)
        queue = [Solution.start_node]
        visited = set()
        count = 0

        while queue:
            for i in range(len(queue)):
                node = queue.pop(0)
                visited.add(node.val)

                if node.left and node.left.val not in visited:
                    queue.append(node.left)
                if node.right and node.right.val not in visited:
                    queue.append(node.right)
                if node.parent and node.parent.val not in visited:
                    queue.append(node.parent)

            if len(queue):
                count += 1

        return count
        
    def convert_to_BI_tree_and_find_start_node(self, bi_node, node, start):
        if not node:
            return None

        if not bi_node:
            bi_node = BI_TreeNode(node.val, node.left, node.right)
        else:
            bi_node.val = node.val
            bi_node.left = node.left
            bi_node.right = node.right

        if node.left:
            bi_node.left = BI_TreeNode(None, None, None, bi_node)
            self.convert_to_BI_tree_and_find_start_node(bi_node.left, node.left, start)
        
        if node.right:
            bi_node.right = BI_TreeNode(None, None, None, bi_node)
            self.convert_to_BI_tree_and_find_start_node(bi_node.right, node.right, start)

        
        if node.val == start:
            Solution.start_node = bi_node

        return bi_node
546