Select the correct alternative from the given choices. A...

Select the correct alternative from the given choices. A scheme for storing binary trees in an array $X$ is as follows: Indexing of $X$ starts at 1 instead of 0 . The root is stored at $X[1]$. For a node stored at $X[i]$, the left child, if any, is stored in $X[2 i]$ and the right child, if any, in $X[2 i+$ 1]. To store any binary tree on ' $n$ ' vertices the minimum size of $X$ should be (A) $2 n$ (B) $n$ (C) $3 n$ (D) $n^{2}$

Select the correct alternative from the given choices.
A scheme for storing binary trees in an array $X$ is as follows:
Indexing of $X$ starts at 1 instead of 0 . The root is stored at $X[1]$. For a node stored at $X[i]$, the left child, if any, is stored in $X[2 i]$ and the right child, if any, in $X[2 i+$ 1]. To store any binary tree on ' $n$ ' vertices the minimum size of $X$ should be
(A) $2 n$
(B) $n$
(C) $3 n$
(D) $n^{2}$

Key Concepts

Array Representation of Binary Trees

This concept refers to the method of storing a binary tree in an array where each element of the tree is placed in a specific index of an array. The common scheme involves starting the indexing at 1, placing the root at index 1, and then for a node stored at index i, placing the left child at index 2i and the right child at index 2i+1. This method is typically used in implementations of complete binary trees, such as heaps, but it can be applied to arbitrary binary trees as well with the understanding that some positions in the array may remain unused.

Indexing Scheme and Its Implications

Understanding the indexing scheme (i.e. using formulas like 2i for the left child and 2i+1 for the right child) is critical because it establishes a direct relationship between the position of a node in the tree and its corresponding index in the array. This relationship implies that even if the tree is not complete, the array may need to be larger than the total number of nodes in order to accommodate the largest index used by any node. Thus, a poor layout (for instance, a tree that is highly skewed) can lead to a requirement for a larger array even though the actual number of nodes is small.

Worst?Case Storage Requirement

For any binary tree with n nodes stored according to the described array representation, the worst-case scenario for the maximum index used can be shown to be less than 2n. This means that in order to be able to store any binary tree on n vertices, the minimum required size of the array is on the order of 2n. This worst-case analysis is important because it guarantees that regardless of the tree’s shape (whether complete or highly skewed), an array sized appropriately (approximately 2n) will always be sufficient, making option (A) a correct answer.

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