Question

Design a function to check if a binary tree is balanced. A balanced tree is defined to be a tree such that the heights of the two subtrees of any node never differ by more than one.

Short answer

Use a recursive function to check subtree balance and propagate results up the tree.

Step-by-step solution

Understand the Problem Statement

A binary tree is balanced if, for every node in the tree, the difference in height between its left and right subtrees is at most 1. Our task is to design a function that returns true if a given binary tree is balanced and false otherwise.

Plan the Solution

We will write a recursive function that checks the balance of a tree. For each node, the function will determine the heights of the left and right subtrees, and check if the difference is greater than 1. If any subtree is unbalanced, the function should immediately return false. Otherwise, it moves up to the parent node.

Define the Recursive Function

Let's define our recursive function called `checkHeight`. This function will return two values: whether the subtree is balanced and its height. The function will return (-1) indicating an unbalanced subtree. We'll use postorder traversal to ensure we check both subtrees before checking the root.

Implement the Base Case

The base case of our recursive function should handle the scenario when the node is `null` (an empty subtree), which is inherently balanced with height 0.

Implement Recursive Calls

For each node, recursively call the function on its left and right children to determine the height of each subtree. If either subtree is unbalanced (returns -1), propagate this result upward by returning -1.

Check Balance Condition

For the current node, check if the difference in heights between the left and right subtrees is greater than 1. If it is, return -1 to indicate the tree is unbalanced. Otherwise, calculate the height at this node.

Return the Result

In the main function that initially called the recursive function, check if the result is -1. If it is, return false. Otherwise, return true, indicating the tree is balanced.

Key concepts

Tree Traversal

Tree traversal is a critical concept in dealing with binary trees. In the problem of checking whether a binary tree is balanced, we often perform tree traversal to examine every node.

• Traversal Types: The primary types of tree traversal are Preorder, Inorder, Postorder, and Level-order. Each serves a different use. In our solution, we are interested in Postorder traversal.
• Why Postorder: Postorder traversal processes the left and right children of a node before the node itself. This is essential for balanced tree checking because we first need to determine the heights of both subtrees before checking if their balance condition holds for the root node.
• Postorder Mechanics: By traversing Postorder, we ensure that our solution calculates the subtree's properties needed for deciding the node's balance status.

Postorder traversal navigates from the bottom to the top, ensuring every subtree is processed entirely before considering its impact on the parent node. This approach is particularly beneficial for recursive solutions that depend on the evaluation of children nodes first.

Recursion in Programming

Recursion is a powerful tool in programming that allows a function to call itself to solve smaller subproblems. In checking if a tree is balanced, recursion provides an elegant way to structure our solution.

• Recursive Structure: The recursive function is designed to break down the problem into manageable parts. It checks each node to determine the balance of its subtrees.
• Base Case: All recursive functions need a base case to stop further recursive calls. In our scenario, a `null` node is considered balanced with a height of 0, serving as the base case.
• Recursive Calls: Each function call will process a node and use recursion to determine the height and balance of its left and right subtrees. The results of these recursive calls are then used to determine the node's balance status and report it upward.

The recursive nature simplifies complex problems by tackling each component separately, collecting and propagating balance information as it recurses up the tree.

Data Structures

Understanding data structures is key when working with binary trees and similar problems. A binary tree is a classic data structure where each node can have at most two children.

• Binary Tree Properties: Nodes of a binary tree are structured with a value, a reference to the left child, and a reference to the right child.
• Balanced Tree Characteristics: In a balanced binary tree, the depth of leaf nodes is not vastly different. This ensures minimal height, optimizing operations like insertion and search.
• Practical Importance: Balancing a binary tree results in efficient operations. It ensures that tree traversal takes the least time, making algorithms like searching and insertion more speedy.

In conclusion, grasping how different data structures work, and utilizing them effectively, is crucial. Whether it's ensuring operations on a binary tree are efficient or understanding how tree nodes are connected, the fundamentals of data structures guide these processes.