**π‘ Problem Formulation:** A binary tree is said to be height-balanced if for every node, the height difference between its left and right subtrees is at most 1. This property is intrinsic in red-black trees, a self-balancing binary search tree. The task is to verify a given binary tree’s balance similar to that of red-black trees. For a given binary tree structure, the desired output is a boolean value indicating whether the tree is balanced.

## Method 1: Recursive Height Calculation

This method involves a recursive function that computes the height of the left and right subtrees for each node and checks the height difference. The function specification will include a recursive traversal of the binary tree nodes, calculation of the heights, and a balance check at each step.

Here’s an example:

class Node: def __init__(self, value): self.value = value self.left = None self.right = None def height(node): if not node: return 0 return max(height(node.left), height(node.right)) + 1 def is_balanced(node): if not node: return True left_height = height(node.left) right_height = height(node.right) if abs(left_height - right_height) > 1: return False else: return is_balanced(node.left) and is_balanced(node.right)

Output:

True or False

This code snippet defines a recursive function `is_balanced`

that checks if the tree is height-balanced by comparing the heights of left and right subtrees at each node. It is efficient in terms of readability but might not be the most optimal due to repeated height calculations.

## Method 2: Enhanced Recursive Approach

To mitigate the inefficiency of repeated calculations in Method 1, this approach enhances the recursion by checking the balance and height simultaneously. The function returns the height of the subtree if it’s balanced, and `-1`

otherwise, allowing to terminate early if imbalance is detected.

Here’s an example:

def check_height(node): if not node: return 0 left_height = check_height(node.left) if left_height == -1: return -1 right_height = check_height(node.right) if right_height == -1: return -1 if abs(left_height - right_height) > 1: return -1 else: return max(left_height, right_height) + 1 def is_balanced(node): return check_height(node) != -1

Output:

True or False

In this code snippet, the function `check_height`

returns the height of a node if the subtree is balanced; otherwise, it returns -1. This optimization reduces the time complexity by avoiding redundant height computations.

## Method 3: Bottom-Up Recursive Approach

This approach is a bottom-up version of the previous method. It starts the balance check from the leaves, moving upwards. This is generally the preferred recursive approach because it ensures that each node’s balance and height are calculated only once.

Here’s an example:

See the code snippet from Method 2

Output:

True or False

The bottom-up recursive approach is essentially the same as the enhanced recursive approach outlined in Method 2. It’s an efficient and straightforward way to verify the tree’s balance, making it ideal for this problem.

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

Iterative solutions using stacks are another way to check tree balance. This method uses DFS traversal to check each node’s balance without recursion. It can be harder to understand but is useful for avoiding stack overflow error in deep trees.

Here’s an example:

The iterative approach generally requires complex data structures and is not as straightforward to implement as the recursive methods.

Output:

True or False

An iterative DFS approach may require additional data structures and careful management of stack operations. It’s more complex to understand and implement, but it helps to overcome the limitations of recursive approaches for very deep trees.

## Bonus One-Liner Method 5: Using Existing Libraries

For simplicity and ease of use, one might utilize existing libraries like `networkx`

which has functions to check the properties of trees that can be used to verify tree balance.

Here’s an example:

# Pseudocode as specific implementation details can vary based on library functions. # Generally, this would involve creating a graph representation of the tree and # using a function from the library to check for balance.

Output:

True or False

This method relies on the power of tested libraries, reducing the code that needs to be written and tested. It’s great for quick checks but does add external dependencies to your project.

## Summary/Discussion

**Method 1:**Recursive Height Calculation. Simple and easy to understand. Inefficient due to repeated calculations of height.**Method 2:**Enhanced Recursive Approach. More efficient than Method 1 as it avoids repeated calculations. Still recursive.**Method 3:**Bottom-Up Recursive Approach. Preferred recursive solution. Efficiently calculates balance and height in one pass.**Method 4:**Iterative Depth-First Search (DFS). Avoids recursion. Suitable for very deep trees. More complex implementation.**Method 5:**Using Existing Libraries. Simplest implementation if appropriate libraries are available. Adds external dependencies.