11from collections import deque
2+ 23def symmetric_tree (root ):
4+ """
5+ Determine if a binary tree is symmetric around its center.
6+
7+ Parameters:
8+ root (Node): The root node of the binary tree.
9+
10+ Returns:
11+ bool: True if the tree is symmetric, False otherwise.
12+ """
313 queue = deque ([root .left , root .right ])
414
515 while queue :
616 curr_left = queue .popleft ()
717 curr_right = queue .popleft ()
818
19+ # If both nodes are None, they are symmetric; continue to the next pair
920 if not curr_left and not curr_right :
1021 continue
1122
23+ # If only one of the nodes is None, the tree is not symmetric
1224 if not curr_right or not curr_left :
1325 return False
1426
27+ # If the values of the nodes are not equal, the tree is not symmetric
1528 if curr_left .val != curr_right .val :
1629 return False
1730
31+ # Append children in mirrored order to maintain symmetry check
1832 queue .append (curr_left .left )
1933 queue .append (curr_right .right )
2034 queue .append (curr_left .right )
2135 queue .append (curr_right .left )
2236
23- return True
37+ return True
38+ 39+ # Approach:
40+ # ---------
41+ # The function uses an iterative approach with a queue to perform a level-order traversal,
42+ # comparing nodes in pairs to check for symmetry. The idea is to compare the left and right
43+ # subtrees of the tree simultaneously, ensuring that they mirror each other.
44+ 45+ # Steps:
46+ # 1. Initialize a queue with the left and right children of the root.
47+ # 2. While the queue is not empty, pop two nodes at a time (curr_left and curr_right).
48+ # 3. If both nodes are None, continue to the next pair (they are symmetric).
49+ # 4. If only one of the nodes is None, return False (they are not symmetric).
50+ # 5. If the values of the nodes are not equal, return False (they are not symmetric).
51+ # 6. Append the children of curr_left and curr_right to the queue in a mirrored order:
52+ # - Append curr_left.left and curr_right.right
53+ # - Append curr_left.right and curr_right.left
54+ # 7. If all pairs are symmetric, return True.
55+ 56+ # Time Complexity:
57+ # ----------------
58+ # O(n), where n is the number of nodes in the tree. Each node is enqueued and dequeued once.
59+ 60+ # Space Complexity:
61+ # -----------------
62+ # O(n), where n is the number of nodes in the tree. In the worst case, the space used by the queue
63+ # is proportional to the number of nodes at the widest level of the tree, which can be up to n/2.
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