Define the largest gap in a collection of numbers (also known as its range) to be greatest difference between two elements in the collection (in absolute value). For example, the largest gap in \(\{4, 9, 5, 1, 6\}\) is 8 (between 1 and 9).

Suppose a collection of \(n\) numbers is stored in a balanced binary search tree. What is the time complexity required of an efficient algorithm to calculate the largest gap in the collection? State your answer as a function of \(n\) in asymptotic notation.

Suppose the numbers 41, 32, and 40 are inserted (in that order) into the below binary search tree. Draw where the new nodes will appear.

Define the smallest gap in a collection of numbers to be the smallest difference between two distinct elements in the collection (in absolute value). For example, the smallest gap in \(\{4, 9, 1, 6\}\) is 2 (between 4 and 6).

Suppose a collection of \(n\) numbers is stored in a balanced binary search tree. What is the time complexity required for an efficient algorithm to calculate the smallest gap of the numbers in the BST? State your answer as a function of \(n\) in asymptotic notation.

Suppose the numbers 41, 32, and 33 are inserted (in that order) into the below binary search tree. Draw where the new nodes will appear.

Define the largest gap in a collection of numbers to be the largest difference between two distinct elements in the collection (in absolute value). For example, the largest gap in \(\{4, 9, 1, 6\}\) is 8 (between 1 and 9).

Suppose a collection of \(n\) numbers is stored in a balanced binary search tree. What is the time complexity required for an efficient algorithm to calculate the largest gap of the numbers in the BST? State your answer as a function of \(n\) in asymptotic notation.

Suppose a collection of \(n\) numbers is stored in a balanced binary search tree. What is the time complexity required of an efficient algorithm to calculate the mean of the collection? State your answer as a function of \(n\) in asymptotic notation.