Height and Diameter in Binary Tree in C++
A binary tree is a data structure that is composed of nodes. Each node has a value, and it can have two children: a left child and a right child. In this article, we will discuss two important concepts in binary trees: height and diameter. We will also discuss how to implement these concepts in C++.
Height of a Binary Tree
The height of a binary tree is the number of edges on the longest path from the root node to a leaf node. In other words, it is the maximum depth of the tree. For example, in the binary tree shown below, the height is 3.
1
/ \
2 3
/ \ \
4 5 6
The height of a binary tree can be calculated using a recursive algorithm. The base case is when the tree is empty, in which case the height is -1. If the tree is not empty, then the height is the maximum of the heights of the left and right subtrees plus 1. The following C++ function calculates the height of a binary tree:
int height(Node *root) {
if (root == nullptr) return -1;
return max(height(root->left), height(root->right)) + 1;
}
Diameter of a Binary Tree
The diameter of a binary tree is the length of the longest path between any two nodes in the tree. This path may or may not pass through the root. For example, in the binary tree shown below, the diameter is 5.
1
/ \
2 3
/ \ \
4 5 6
The diameter of a binary tree can be calculated using a recursive algorithm. The base case is when the tree is empty, in which case the diameter is 0. If the tree is not empty, then the diameter is the maximum of the following:
- The diameter of the left subtree
- The diameter of the right subtree
- The length of the path that goes through the root (left height + right height + 1)
The following C++ function calculates the diameter of a binary tree:
int diameter(Node *root) {
if (root == nullptr) return 0;
int leftHeight = height(root->left);
int rightHeight = height(root->right);
int leftDiameter = diameter(root->left);
int rightDiameter = diameter(root->right);
return max({leftDiameter, rightDiameter, leftHeight + rightHeight + 1});
}
It is important to note that the above implementation has a time complexity of O(n^2) since the height function is called twice for each node. This can be optimized to O(n) by calculating the height and diameter simultaneously in the same function.
int diameterOptimized(Node *root, int &height) {
if (root == nullptr) {
height = -1;
return 0;
}
int leftHeight = 0, rightHeight = 0;
int leftDiameter = diameterOptimized(root->left, leftHeight);
int rightDiameter = diameterOptimized(root->right, rightHeight);
height = max(leftHeight, rightHeight) + 1;
return max({left
Diameter, rightDiameter, leftHeight + rightHeight + 1}); }
In this optimized version, the height of the tree is passed as a reference parameter to the function and is updated during the recursive call. This eliminates the need to call the height function separately, reducing the overall time complexity to O(n).
Conclusion
In this article, we have discussed the concepts of height and diameter in binary trees and how to implement them in C++. The height of a binary tree is the number of edges on the longest path from the root node to a leaf node. The diameter of a binary tree is the length of the longest path between any two nodes in the tree. Both of these concepts can be calculated using a recursive algorithm, with the height calculation being a simpler version of the diameter calculation. The time complexity of the diameter calculation can be optimized from O(n^2) to O(n) by calculating the height and diameter simultaneously in the same function. By understanding and implementing these concepts, we can gain a deeper understanding of binary trees and their properties.
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