If a hash table is implemented as a search tree, the expected time required to enter n names and make m searches is proportional to : 

The correct answer is (n + m) log₂ n.

Key Points

• If a hash table is implemented as a search tree, the expected time required to enter n names and make m searches is proportional to (n + m) log₂ n.
• The reason behind this is that the search tree ensures that the time complexity for insertion and search operations is logarithmic relative to the number of elements in the tree.
• Inserting n names takes O(n log₂ n) time, as each insertion can take up to log₂ n comparisons in a balanced tree.
• Searching for m names takes O(m log₂ n) time, as each search operation can take up to log₂ n comparisons.
• Therefore, the total expected time is the sum of the time for insertion and search operations, which is (n + m) log₂ n.

Additional Information

• A search tree, such as a binary search tree (BST), AVL tree, or Red-Black tree, is used to maintain a sorted order of elements, allowing for efficient search and insertion operations.
• The balancing property of search trees ensures that the height of the tree remains logarithmic relative to the number of elements, thus providing efficient operations.
• Hash tables are typically used for average-case constant-time complexity operations, but when implemented as search trees, they leverage the tree's logarithmic properties for better worst-case performance.
• This implementation is useful in scenarios where the data set is dynamic and needs to be frequently updated and searched efficiently.