$1.$ Develop an algorithm that can be used to determine whether a Stack object S has exactly one element.

$2.$ Develop an algorithm that adds the value val to a queue object Q$.$ The queue is represented using an array data of size s$.$ The algorithm should assume that the queue is not full. The most recently added item is at index r $($rear$),$ and the least recently added item is at index f $($front$).$ If the queue is empty, r $=$ f $=-1.$

$3.$ Develop an algorithm that removes the least recently added item from a queue Q$.$ The queue is represented using an array of size s$.$ The algorithm assumes that the queue is not empty. The most recently added item is at index r $($rear$),$ and the least recently added item is at index f $($front$).$ If the queue is empty, r $=$ f $=-1.$

$4.$ Develop an algorithm that computes the degree of each vertex in a graph. The graph is represented using adjacency lists A; A$[$i$]$ is a reference to the first node in a linked list of nodes representing the vertices adjacent to vertex i$.$ The next field in each node, except the last, references the next node in the list. The next field of the last node is null. The vertices are $1,2,....$

$5.$ Develop a recursive algorithm that returns the number of nodes in a binary tree with root r$.$

$6.$ Develop an algorithm that inserts the value val into a binary search tree with root. If the tree is empty, root $=$ null. The algorithm returns the root of the tree containing the added item. You should assume that $$new node$$ creates a new node with data field data and reference fields left $($for left child$)$ and right $($for right child$).$

$7.$ Show each AVL tree created from each of the following insertion: $3,2,1,4,5,6,7,16,15,14$

$8.$ Show the AVL tree created by the following array $\{20,30,80,40,10,60,50,70\}$

$9.$ Given the following Splay tree, show the resulting tree after removing the node labeled $6.$

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$10.$ Given the following B$-$tree of order $5,$ show the resulting trees after each of the following operations: $($a$)$ Insert $12($b$)$ Insert $53.$

$11.$ Prove that if n keys are uniformly distributed in m buckets, the number of comparisons in an unsuccessful search is given by n$/$m$.$

$12.$ Show the heap built from this array $\{150804030107011010020906050120140130\}.$ Then fix the heap to become a priority queue tree.