Answer the following questions

$1)$ Draw a binary balanced search tree without a balancing rotation mechanism for the following values, and explain your strategy to build such sorted n number of elements:

$431231853434114558349411310$

$2)$ Draw a new binary search tree that results from adding the following integers $(14,214,33,918,15,48,56,815,134,10,131,125,45).$ Assume our simple implementation with no balancing mechanism.

$3)$ Starting with the result from Exercise $2,$ draw the tree that results from removing $(8153310,15),$ again using our simple implementation with no balancing mechanism.

$4)$ Draw a new AVL Balanced Tree that results from adding the following integers $(14,214,33,918,15,48,56,815,134,10,131,125,45)$ and show each iteration.

$5)$ Starting with the resulting tree from Exercise $4,$ draw the tree that results from performing a remove the root operation and show each iteration according to AVLTree remove process $($remove root only one time$).$

$6)$ Write a method to take a BinayTreeNode input $($root$)$ for level order, preOrder, postOrder, and inOrder methods.

$7)$ Write a method to take a BinayTreeNode input $($root$)$ and determine whether the given tree is Binary Search Tree

$8)$ Suppose you are given the following set of keys $(111,114,118,95,99,113,128,117,115,99)$ to insert into a hash table that holds exactly $11$ values. Which index positions would those keys map to according to chaining?

$9)$ Suppose you are given the following set of keys $(111,114,118,95,99,113,128,117,115,99)$ to insert into a hash table that holds exactly $11$ values. Which index positions would those keys map to according to linear probing?

$10)$ Suppose you are given the following set of keys $(111,114,118,95,99,113,128,117,115,99)$ to insert into a hash table that holds exactly $11$ values. Which index positions would those keys map to according to quadratic probing?

$11)$ Suppose you are given the following set of keys $(111,114,118,95,99,113,128,117,115,99)$ to insert into a hash table that holds exactly $11$ values. Which index positions would those keys map to according to h $=($x $+$ i$*(7$ x mod $7))$ mod $11?$

$12)$ Draw the minHeap that results from adding the following integers $(34,40,58,451008765321167).$

$13)$ Starting with the resulting tree from Exercise $12,$ draw the minHeap that results from performing a remove root operation.

$14)$ Repeat Exercise $12$ by using the same numbers to create a new maxHeap and remove the root from the maxHeap.

$15)$ Draw the undirected graph that is represented by the following:

vertices: $1,2,3,4,5,6,7$

edges: $(1,6),(1,2),(2,4),(2,7),(3,5),(4,3),(4,5),(6,3),(5,7),(6,7),(3,2)$

$16)$ Write a matrix representation of the Exercise $15$ graph.

$17)$ Is the graph from Exercise $15$ connected? Is it complete?

$18)$ List all of the cycles in the graph from Exercise $15.$

$19)$ Use the resulting graph of Exercise $15.$ Starting from vertex $1$ and apply breadth$-$first search traversal by visiting the vertex according to increasing order.

$20)$ Use the resulting graph of Exercise $15.$ Starting from vertex $1$ and apply depth$-$first search traversal by visiting the vertex according to increasing order.

$21)$ Draw a spanning tree for the graph of Exercise $15.$

$22)$ Using the same data from Exercise $15,$ draw the resulting directed graph.

$23)$ Is the directed graph of Exercise $22$ connected? Is it complete?