Question

For the set of keys 5, 49, 51, 62, 84, 99, 108, draw binary search trees of heights 2, 3, 4, 5, and 6, respectively. Note that you can pick any key as the root. The height of a tree is the number of edges on the longest simple downward path from the root to a leaf.

Answer 1

To create binary search trees with heights ranging from 2 to 6 for the given set of keys, distinct configurations must be made by choosing appropriate roots and placing children to maintain the tree height.

Step-by-step explanation:

The question involves drawing binary search trees (BSTs) of various heights for a given set of keys. The keys provided are 5, 49, 51, 62, 84, 99, and 108. A binary search tree is a node-based data structure where each node has at most two children referred to as the left child and the right child. The left subtree of a node contains only nodes with keys less than the node's key, and the right subtree only contains nodes with keys greater than the node's key. To ensure a specific height for the binary search tree, the root must be chosen carefully, and subsequent children must be placed to maintain the requested tree height.

Here are examples for the required heights:

• Height 2: Choose 62 as the root, with 49 as left child and 84 as right child. Then, 5 under 49 and 51 as its right child, and 99 under 84 with 108 as its right child.
• Height 3: Choose 51 as the root, with 49 to the left, and 62 to the right. Then, 5 under 49, and 84 under 62. Finally, add 99 as the right child of 84, with 108 to its right.
• Height 4: Choose 49 as the root, with 5 to the left, and 51 to the right. Branching from 51, place 62 to the right, then 84 to the right of 62, followed by 99, and finally 108 to the right of 99.
• Height 5: Choose 49 as the root with 5 to the left and 62 to the right. Under 62, place 51 to the left and 84 to the right, then continue with 99 to the right of 84, and end with 108 to the right of 99.
• Height 6: Choose 5 as the root, then place 49 to the right. Next, 51 to the right of 49, followed by 62 to the right of 51, then 84, 99, and finally 108, each to the right of its predecessor.

These binary search trees are only examples; other valid configurations will exist as long as the tree structure satisfies the BST properties and the requirement of having a specific height.