Final Answer:
The number of distinct paths from the root to each external node at the last level of a complete binary tree with height H and N nodes is 2^H.
Explanation:
In a complete binary tree, each internal node has two children, a left child, and a right child. The height of the tree, denoted as H, represents the number of levels in the tree. The last level of the tree contains the external nodes. Each path from the root to an external node can be represented by an H-tuple of 1s and 0s, where 0 indicates a left child and 1 indicates a right child.
To calculate the number of distinct paths, we consider that at each level, there is a choice of two directions (left or right) to reach the next level. Since there are H levels in total, the total number of distinct paths is 2 * 2 * ... * 2 (H times), which simplifies to 2^H. This is because each decision to go left or right is independent of the previous choices, resulting in an exponential growth of possibilities.
In summary, the total number of distinct paths from the root to each external node at the last level of a complete binary tree with height H is given by 2^H. This exponential growth is a fundamental property of complete binary trees and is crucial for understanding their structural characteristics and traversal algorithms.