<p>Given a binary tree <code>root</code> and a&nbsp;linked list with&nbsp;<code>head</code>&nbsp;as the first node.&nbsp;</p>

<p>Return True if all the elements in the linked list starting from the <code>head</code> correspond to some <em>downward path</em> connected in the binary tree&nbsp;otherwise return False.</p>

<p>In this context downward path means a path that starts at some node and goes downwards.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><strong><img alt="" src="https://assets.leetcode.com/uploads/2020/02/12/sample_1_1720.png" style="width: 220px; height: 280px;" /></strong></p>

<pre>
<strong>Input:</strong> head = [4,2,8], root = [1,4,4,null,2,2,null,1,null,6,8,null,null,null,null,1,3]
<strong>Output:</strong> true
<strong>Explanation:</strong> Nodes in blue form a subpath in the binary Tree.  
</pre>

<p><strong class="example">Example 2:</strong></p>

<p><strong><img alt="" src="https://assets.leetcode.com/uploads/2020/02/12/sample_2_1720.png" style="width: 220px; height: 280px;" /></strong></p>

<pre>
<strong>Input:</strong> head = [1,4,2,6], root = [1,4,4,null,2,2,null,1,null,6,8,null,null,null,null,1,3]
<strong>Output:</strong> true
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> head = [1,4,2,6,8], root = [1,4,4,null,2,2,null,1,null,6,8,null,null,null,null,1,3]
<strong>Output:</strong> false
<strong>Explanation:</strong> There is no path in the binary tree that contains all the elements of the linked list from <code>head</code>.
</pre>

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

<ul>
	<li>The number of nodes in the tree will be in the range <code>[1, 2500]</code>.</li>
	<li>The number of nodes in the list will be in the range <code>[1, 100]</code>.</li>
	<li><code>1 &lt;= Node.val&nbsp;&lt;= 100</code>&nbsp;for each node in the linked list and binary tree.</li>
</ul>

# Definition for singly-linked list.
# class ListNode:
#     def __init__(self, val=0, next=None):
#         self.val = val
#         self.next = next
# 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 Solution:
    
    def isSubPath(
        self, head: Optional[ListNode], root: Optional[TreeNode]
    ) -> bool:

        # Build the pattern and prefix table from the linked list
        pattern = [head.val]
        prefix_table = [0]
        pattern_index = 0
        head = head.next

        while head:
            while pattern_index > 0 and head.val != pattern[pattern_index]:
                pattern_index = prefix_table[pattern_index - 1]
            pattern_index += 1 if head.val == pattern[pattern_index] else 0
            pattern.append(head.val)
            prefix_table.append(pattern_index)
            head = head.next

        print(prefix_table)
        print(pattern)

        # Perform DFS to search for the pattern in the tree
        return self._search_in_tree(root, 0, pattern, prefix_table)

    def _search_in_tree(
        self,
        node: Optional[TreeNode],
        pattern_index: int,
        pattern: List[int],
        prefix_table: List[int],
    ) -> bool:
        if not node:
            return False

        while pattern_index > 0 and node.val != pattern[pattern_index]:
            pattern_index = prefix_table[pattern_index - 1]
        pattern_index += 1 if node.val == pattern[pattern_index] else 0

        # Check if the pattern is fully matched
        if pattern_index == len(pattern):
            return True

        # Recursively search left and right subtrees
        return self._search_in_tree(
            node.left, pattern_index, pattern, prefix_table
        ) or self._search_in_tree(
            node.right, pattern_index, pattern, prefix_table
        )

        
        
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