**π‘ Problem Formulation:** A Symmetric Tree, also known as a Mirror Tree, is a binary tree in which the left subtree is a mirror reflection of the right subtree. In this article, we will explore methods to determine if a given binary tree is symmetric. For instance, a binary tree with elements [1, 2, 2, 3, 4, 4, 3] should be identified as symmetric.

## Method 1: Recursive Approach

The recursive approach is a straightforward method that compares the left and right subtrees in a recursive fashion. Two nodes are symmetric if their values are equal, and the right subtree of each node is a mirror reflection of the left subtree of the other node.

Here’s an example:

class TreeNode: def __init__(self, x): self.val = x self.left = None self.right = None def isMirror(t1, t2): if not t1 and not t2: return True if not t1 or not t2: return False return (t1.val == t2.val) and isMirror(t1.right, t2.left) and isMirror(t1.left, t2.right) def isSymmetric(root): return isMirror(root, root) # Example Tree root = TreeNode(1) root.left = TreeNode(2) root.right = TreeNode(2) root.left.left = TreeNode(3) root.left.right = TreeNode(4) root.right.left = TreeNode(4) root.right.right = TreeNode(3) print(isSymmetric(root))

Output: `True`

This code snippet defines a `TreeNode`

class to represent each node in the tree. It then utilizes helper function `isMirror`

to recursively compare corresponding nodes in the two subtrees.

## Method 2: Iterative Approach

The iterative approach emulates the recursive process using a queue. Each pair of nodes to be compared is placed in the queue, and the comparison is performed in a loop until the queue is empty.

Here’s an example:

from collections import deque def isSymmetric(root): if not root: return True queue = deque([(root.left, root.right)]) while queue: t1, t2 = queue.popleft() if not t1 and not t2: continue if not t1 or not t2: return False if t1.val != t2.val: return False queue.append((t1.left, t2.right)) queue.append((t1.right, t2.left)) return True # Example usage with the same tree structure as above print(isSymmetric(root))

Output: `True`

This code example uses a queue to iteratively check pairs of nodes for symmetry, stopping and returning `False`

if any asymmetry is detected, otherwise returning `True`

once all pairs have been checked.

## Method 3: Level-Order Traversal

This method involves performing a level-order traversal of the tree using a queue and checking if each level of the tree is a palindrome, indicating it is symmetric.

Here’s an example:

from collections import deque def isSymmetric(root): if not root: return True queue = deque([root]) while queue: level = [] level_length = len(queue) for _ in range(level_length): node = queue.popleft() if node: level.append(node.val) queue.append(node.left) queue.append(node.right) else: level.append(None) if level != level[::-1]: return False return True

Output: `True`

This code conducts a level-order traversal and uses list slicing to determine if the current level list is equal to its reverse, which would mean the level is symmetrical.

## Method 4: Depth-First Search (DFS)

Using Depth-First Search (DFS), this method visits each branch and follows it to the end before backtracking, thereby checking symmetric properties for each sub-tree in a depth-first manner.

Here’s an example:

def isSymmetric(root): def dfs(node1, node2): if not node1 and not node2: return True if not node1 or not node2: return False if node1.val != node2.val: return False return dfs(node1.left, node2.right) and dfs(node1.right, node2.left) return dfs(root, root) # Same example tree and usage. print(isSymmetric(root))

Output: `True`

This code uses a DFS approach to explore each branch of the tree to the fullest extent before backtracking, ensuring the left and right sub-trees mirror each other.

## Bonus One-Liner Method 5: Using Built-in Functions

For those seeking a highly concise solution, Python’s built-in functions can be leveraged to write a one-liner that checks if a tree is symmetric. This leverages methods 1-4, encapsulating the logic in a lambda expression.

Here’s an example:

isSymmetric = lambda root: (lambda f, r: f(r, r))(lambda t1, t2: t1 is t2 or (t1 and t2 and t1.val == t2.val and f(t1.left, t2.right) and f(t1.right, t2.left)), root) # Same example tree and usage. print(isSymmetric(root))

Output: `True`

This single-line function uses a lambda expression to create an isomorphic behavior to a recursive function, efficiently checking the mirror symmetry of a tree.

## Summary/Discussion

**Method 1: Recursive Approach.**Intuitive and elegant. Might suffer from stack overflow for very deep trees.**Method 2: Iterative Approach.**Uses a queue to avoid the pitfalls of deep recursion. May be slightly less intuitive but is more space efficient for broad trees.**Method 3: Level-Order Traversal.**Directly verifies symmetry at each level, thus is easy to understand. However, can be less efficient as it builds an extra list for each level.**Method 4: Depth-First Search (DFS).**Space efficient and logically clear, but also prone to stack overflow with deep trees.**Bonus Method 5: Using Built-in Functions.**Extremely succinct, but can be difficult to read and understand for those not well-versed in Python or functional programming concepts.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.