Problem $1$: For each array of elements given below, identify whether or not the array is a heap. If not, explain in detail why the array cannot be a heap.

Array $1$ :

$37914821610232011793182111$

Array $2$ :

$51310422011124045262818146$

Problem $2$: Assume we have a heap that guarantees the smallest element is always at the top and that the heap is initially empty. Draw a diagram of the heap that is constructed when the following integers are added to the heap, one by one, in the order given.

$14826119467218116317654$

Then, draw the heap after the smallest element is removed.

Problem $3$: Write a recursive method in Java that prints a Binary Search Tree $($BST$)$ in reverse order. $($For integers, this means you are printing the number in descending order.$)$

Assume you are writing the method in the BST class and have access to the Node class with the fields: element, left and right.

Problem $4$: For each of the following sequences of numbers:

Draw the BST that is created if you add the numbers one at a time in the order given.

Identify whether the completed BST is also an AdelsonVelsky Landis $($AVL$)$ tree.

If the BST is not an AVL tree, identify all unbalanced nodes and explain why each is unbalanced.

a$)52867314$

b$)41739856$

c$)69415328$

Problem $5$: Clearly explain the benefits of keeping a BST balanced. Be specific.

Problem $6$: For each of the following sequences of numbers:

Draw the AVL tree that is created if you add the numbers one at a time in the order given.

Clearly identify each rotation.

a$)327169485$

b$)42139758$

Problem $7$: Consider the following BST:a$)$ Write the pre$-$order sequence of integers.b$)$ Write the Java code that deletes the number $5$ from the tree. Assume there is a Node class with the fields: element, right and left. Also assume there is a root variable the references the $10.$ Do not write the general remove method...just write the code that specifically removes $5$ from this tree.Problem $8$: For each data structure listed below, identify one benefit and one drawback of using the data structure for a particular problem.a$)$ arrayb$)$ linked listc$)$ stackd$)$ heape$)$ hash tablef$)$ binary search tree $($balanced$)$