Consider a binary tree with n nodes, where each node can have at most two children. The height of the tree is defined as the maximum number of edges between the root node and any leaf node. Which of the following statements is true regarding the height h of this binary tree?*The height of the tree is always equal to n-1The height of the tree is always equal to log₂(n)The height of the tree can be greater than or equal to n-1
Question
Consider a binary tree with n nodes, where each node can have at most two children. The height of the tree is defined as the maximum number of edges between the root node and any leaf node. Which of the following statements is true regarding the height h of this binary tree?*The height of the tree is always equal to n-1The height of the tree is always equal to log₂(n)The height of the tree can be greater than or equal to n-1
Solution
None of the statements provided are universally true for all binary trees.
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The height of the tree is always equal to n-1: This is not always true. This would only be true for a skewed binary tree (a tree in which each node only has one child). In this case, the tree essentially becomes a linked list, and the height is indeed n-1. However, in a balanced binary tree, the height is much less than n-1.
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The height of the tree is always equal to log₂(n): This is true only for a perfect binary tree, where all levels are fully filled. In this case, the height is indeed log₂(n+1). However, not all binary trees are perfect binary trees.
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The height of the tree can be greater than or equal to n-1: This is not true. The height of a binary tree can never be greater than n-1. As mentioned above, the height is n-1 only in the case of a skewed binary tree. In all other cases, the height is less than n-1.
So, all of the statements are conditionally true, depending on the specific characteristics of the binary tree.
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