## Question

In this task, you implement two of the various search trees and you should compare the result.

pseudocode (normal):

List-Search(L, k )

1 x = L.head

2 while x != nil and x.key̸ != k

3 x = x.next

4 return x

List-Insert(L,x)

1 x.next = L.head

2 if L.head != nil

3 L.head.prev = x

4 L.head = x

5 x.prev = nil

List-Delete(L, x)

1 if x.prev = nil

2 x.prev.next = x.next

3 else L.head = x.next

4 if x.next = nil

5 x.next.prev = x.prev

List-Delete Sentinels (easier to use sentinels)

List-Delete(L, x)

1 x.prev.next = x.next

2 x.next.prev = x.prev

￼

A - Binary Search Trees

Implement binary search trees under the pseudo code. It maintains that you implement dictionary operations Insert, Delete, and Search.

INORDER-TREE WALK(x)

if x != NIL

INORDER-TREE-WALK(x.left)

print key[x]

INORDER-TREE-WALK(x.right)

Pseudo code for Tree-search algorithm can be used to perform queries on a binary search tree.

TREE-SEARCH

if x == NIL or k == key[x]

return x

if k < x.key

return TREE-SEARCH(x.left, k)

else return TREE-SEARCH(x.right, k)

Property of binary search trees make it very easy to recall the largest and smallest value. Tree-Minimum and Maximum Tree algorithm returns respectively the smallest and largest element in the tree.

TREE-MINIMUM(X)

while x.left != NIL

x = x.left

return x

TREE-MAXIMUM(x)

while x.right != NIL

x = x.right

return x

TREE-INSERT.T; ́/

y = NIL

x = T.root

while x != NIL

y = x

if z.key < x.key

x = x.left

else x = x.right

z.p = y

if y == NIL

T.root = z //tree T was empty

else if z.key < y:key

y.left = z

else y.right = z

TREE-SUCCESSOR(x)

if x.right != NIL

return TREE-MINIMUM(x.right)

y = x.p

while y != NIL and x == y.right

x = y

y = y.p

return y

TRANSPLANT.T; u; /

if u.p == NIL

T.root = v

else if u == u.p.left

u-p-left = v

else u.p.right = v

if v != NIL

v.p = u.p

TREE-DELETE(T, z)

if z.left == NIL

TRANSPLANT(T, z, z.left)

else if z.right == NIL

TRANSPLANT(T, z, z.left)

else // z has twho children

y = TREE-MINIMUM(z.right)

if y.p != z

// y lies within z's right subtree but is not the root of this subtree.

TRANSPLANT(T.y, y.right)

y.right = z.right

y.right.p = y

// Replace z by y.

TRANSPLANT(T, z, y)

y.left = z.left

y.left.p = y

B - Red-Black Trees

Implement red-black trees according to the pseudo code. It also keeps A here that you implement dictionary operations Insert, Delete, and Search.

see the picture of attach file

￼

C - Comparison

In this task implementations is tested against each other. The actual running time for operations to be measured for different input and input sizes. The results will be presented in a table and / or graph, and they will be discussed in the runtime analysis.

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