Lecture 264:- Height vs Nodes in Binary Tree

In a binary tree, the height and the number of nodes are closely related. The height of a binary tree is the maximum depth of the tree, i.e., the length of the longest path from the root node to a leaf node. The number of nodes in a binary tree is the total count of nodes present in the tree.

The relationship between the height and the number of nodes in a binary tree depends on the tree's structure and whether it is a complete binary tree, a full binary tree, or a skewed binary tree.

1. Complete Binary Tree:
   • The height of a complete binary tree with N nodes is log2(N + 1) rounded down to the nearest integer. (Assuming the root node is at level 0)
   • The number of nodes in a complete binary tree with height H is 2^(H+1) - 1.
2. Full Binary Tree:
   • The height of a full binary tree with N nodes is log2(N + 1) rounded down to the nearest integer. (Assuming the root node is at level 0)
   • The number of nodes in a full binary tree with height H is 2^(H+1) - 1.
3. Skewed Binary Tree:
   • In a skewed binary tree, the tree is heavily imbalanced towards one side (left or right).
   • The height of a skewed binary tree with N nodes can be N-1 if the tree is skewed in one direction only.
   • The number of nodes in a skewed binary tree with height H can be as low as H and as high as N, depending on the tree's shape.

In general, the height of a binary tree is related to the logarithm of the number of nodes. For a balanced binary tree, the height is logarithmic with respect to the number of nodes, making tree operations efficient. However, for skewed binary trees, the height can be linear in the number of nodes, which can lead to inefficient operations.

It's important to keep the binary tree balanced to ensure better performance for tree operations like insertion, deletion, and search. Balanced binary trees such as AVL trees or Red-Black trees are designed to maintain a logarithmic height, ensuring efficient operations even for a large number of nodes.