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Chapter 5 Part 2: Search Trees

Binary search trees AVL trees

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Binary Search Trees

All items in the left subtree < the root . All items in the right subtree >= the

root .

Each subtree is itself a binary searchtree.

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Binary Search Tree Traversals

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preorder23 18 12 20 44 3552

postorder12 20 18 35 52 4423

inorder12 18 20 23 35 4452

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Find Smallest Node

Algorithm findSmallestBST (val root )

Finds the smallest node in a BST

Pre root is a pointer to a non-empty BST

Return address of smallest node1 if (root > left = null)

1 return (root)

2 return findSmallestBST (root > left)

End findSmallestBST

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Find Largest Node

Algorithm findLargestBST (val root )

Finds the largest node in a BST

Pre root is a pointer to a non-empty BST

Return address of largest node1 if (root > right = null)

1 return (root)

2 return findLargestBST (root > right)

End findLargestBST

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BST SearchAlgorithm searchBST (val root , val arg )Searches a binary search tree for a given value

Pre root is a pointer to a non-empty BSTarg is the key value requested

Return the node address if the value is found; null otherwise1 if (root = null)

1 return null2 else if (arg = root > key)

1 return root3 else if (arg < root > key)

1 searchBST (root > left, arg)4 else if (arg > root > key)

1 searchBST (root > right, arg)5 else

1 return nullEnd searchBST

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BST Insertion

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Taking place at anode having a nullbranch

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Iterative BST InsertionAlgorithm

Algorithm insertBST (ref root , val new )

Inserts a new node into BST using iteration

Pre root is address of the root

new is address of the new nodePost new node inserted into the tree

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Iterative BST InsertionAlgorithm

1 if (root = null)1 root = new

2 else1 pWalk = root2 loop (pWalk not null)

1 parent = pWalk

2 if (new > key < pWalk > key)1 pWalk = pWalk > left

3 else1 pWalk = pWalk > right

Location found for the new node3 if (new > key < parent > key)

1 parent > left = new4 else

1 parent > right = new3 returnEnd insertBST

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Recursive BST InsertionAlgorithm

Algorithm addBST (ref root , val new )

Inserts a new node into BST using recursion

Pre root is address of the root

new is address of the new nodePost new node inserted into the tree

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Recursive BST InsertionAlgorithm

1 if (root = null)1 root = new

2 else1 if (new > key < root > key)

1 addBST (root > left, new)2 else

1 addBST (root > right, new)3 returnEnd addBST

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BST Deletion

Leaf node : set the deleted node'sparent link to null.

Node having only right subtree : attach

the right subtree to the deletednode's parent.

Node having only left subtree : attach

the left subtree to the deleted node'sparent.

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BST Deletion

Node having both subtrees

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2219

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2219

?

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BST Deletion

Node having both subtrees

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Using largest node in the leftsubtree

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BST Deletion

Node having both subtrees

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Using smallest node in the rightsubtree

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BST Deletion AlgorithmAlgorithm deleteBST (ref root , val dltKey )Deletes a node from a BST

Pre root is a pointer to a non-empty BSTdltKey is the key of the node to be deleted

Post node deleted & memory recycledif dltKey not found, root unchanged

Return true if node deleted; false otherwise1 if (root = null)

1 return false2 if (dltKey < root > data.key)

1 deleteBST (root > left, dltKey)3 else if (dltKey > root > data.key)

1 deleteBST (root > right, dltKey)4 else

Deleted node found

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BST Deletion Algorithm4 else

Deleted node found1 if (root > left = null)

1 dltPtr = root2 root = root > right3 recycle (dltPtr)4 return true

2 else if (root > right = null)1 dltPtr = root2 root = root > left3 recycle (dltPtr)4 return true

3 else1 dltPtr = root > left2 loop (dltPtr > right not null)

1 dltPtr = dltPtr > right3 root > data = dltPtr > data4 return deleteBST (root > left, dltPtr > data.key)

End deleteBST

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AVL Trees

The heights of the left subtree and theright subtree differ by no more thanone.

The left and right subtrees are AVL trees themselves.

(G.M. Adelson- Velskii and E.M. Landis, 1962 )

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AVL Trees

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LH

LH

EH

E

H

EH

EH

EH

EH

EH

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Getting Unbalanced

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insert 4

Left of Left

EH

EH

EH

EH

LH LH

LH

LH

EH

EH

E

H

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Getting Unbalanced

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insert 44

Right of Right

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EH

EH

EH

EH EH RH

EH

EH

RH

RH

RH

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Getting Unbalanced

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insert 13

Right of Left

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LH LH

EH

EH

EH

EH RH

EH

EH

LH

E

H

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Getting Unbalanced

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insert 19

Left of Right

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RH

EH

EHEH

EH

RH

EH LH

RH

EH

EH

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Balancing Trees

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Left of Left

rotate right18

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LH

EH

EH

EH

EH

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Balancing Trees

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insert 4

Left of Left

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rotateright

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LH

EH

EH

EH

EH

EH

LH

EH

EH

EH

EH

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Balancing Trees

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Right of Right

rotate left18

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RH

EH

EH

E

H

EH

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Balancing Trees

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Right of Right

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EH

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Balancing Trees

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Right of Left

rotate right8

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EH

EH

EH

EH

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Balancing Trees

18 rotateleft

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Right of Left

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2014

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rotateright

EH

EH

LH

EH

EH

EH

LH

EH

EH

EH

EH

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Balancing Trees

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rotateright

Left of Right

rotate left18

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RH

LH

EH

EH

EH

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Balancing Trees

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rotateright

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Left of Right

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522012

EH

EH

RH

LH

LH

EH

EH

EH

EH

EH

RH

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AVL Node Structure

Nodekey

data

leftSubTree

rightSubTree bal

End Node

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RH

EH

LH

LH EH

EH

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AVL Insertion Algorithm

Algorithm insertAVL (ref root , val newPtr ,

ref taller )

Inserts a new node into an AVL tree using recursion

Pre root is address of the rootnewPtr is address of the new node

Post taller = true indicating the tree's height has increased;false ortherwise

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AVL Insertion Algorithm

1 if (root = null)1 taller = true2 root = newPtr

2 else1 if (newPtr > key < root > key)

1 insertAVL (root > left, newPtr, taller)

2 if (taller)1 if (root left-high)

1 leftBalance (root, taller)2 else if (root right-high)

1 taller = false2 root > bal = EH

3 else1 root > bal = LH

2 else if (newPtr > key > root > key)

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AVL Insertion Algorithm

2 else if (newPtr > key > root > key)1 insertAVL (root > right, newPtr, taller)2 if (taller)

1 if (root right-high)1 rightBalance (root, taller)

2 else if (root left-high)

1 taller = false2 root > bal = EH

3 else1 root > bal = RH

3 else1 error ("Dupe data")2 recycle (newPtr)3 taller = false

3 returnEnd insertAVL

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AVL Left Balance Algorithm

Algorithm leftBalance (ref root , ref taller )

Balances an AVL tree that is left heavy

Pre root is address of the root

taller = truePost root has been updated (if necessary)

taller has been updated

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AVL Left Balance Algorithm

1 leftTree = root > left2 if (leftTree left-high)

Case 1: Left of left. Single rotation required.1 adjust balance factors2 rotateRight (root)3 taller = false

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EH

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LH

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LH

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LH

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EH

AVL Left Balance Algorithm

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AVL Left Balance Algorithm3 else

Case 2: Right of left. Double rotation required.1 rightTree = leftTree > right2 adjust balance factors3 rotateLeft (leftTree)4 rotateRight (root)5 taller = false

4 return

End leftBalance18

2012

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2014

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EH

EH

E

H

LH

R

H

EH

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LH

EH

EH

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Rotate Algorithms

Algorithm rotateRight (ref root )

Exchanges pointers to rotate the tree right

Pre root is address of the root

Post Node rotated and root updated

Rotate Algorithms

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Rotate Algorithms1 tempPtr = root > left2 root > left = tempPtr > right

3 tempPtr > right = root4 root = tempPtr5 returnEnd rotateRight

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tempPtr20

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root

tempPtr14

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root

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AVL Deletion

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delete 9

Right subtree is even-balanced

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rotateleft

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EH

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LH

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AVL Deletion Algorithm

Algorithm deleteAVL (ref root , val deleteKey ,

ref shorter )

Deletes a node from an AVL tree

Pre root is address of the rootdeleteKey is the key of the node to be deleted

Post node deleted if foundshorter = true if the tree is shorter

Return success true if node deleted; false otherwise

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AVL Deletion Algorithm1 if (root null)

1 shorter = false2 return success = false

2 if (deleteKey < root > key)1 success = deleteAVL (root > left, deleteKey, shorter)2 if (shorter)

1 deleteRightBalance (root, shorter)3 return success

3 else if (deleteKey > root > key)1 success = deleteAVL (root > right, deleteKey, shorter)2 if (shorter)

1 deleteLeftBalance (root, shorter)

3 return success4 else

Delete node found - Test for leaf node

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AVL Deletion Algorithm4 else

Delete node found - Test for node having a null subtree1 deleteNode = root2 if (no left subtree)

1 root = root > right2 shorter = true

3 recycle (deleteNode)4 return success = true

3 else if (no right subtree)1 root = root > left2 shorter = true3 recycle (deleteNode)

4 return success = true4 else

Delete node has two subtrees

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AVL Deletion Algorithm

4 elseDelete node has two subtrees

1 exchPtr = root > left2 loop (exchPtr > right not null)

1 exchPtr = exchPtr > right3 root > data = exchPtr > data4 deleteAVL (root > left, exchPtr > data.key, shorter)5 if (shorter)

1 deleteRightBalance (root, shorter)6 return success = true

5 return

End deleteAVL

Delete Right Balance

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Delete Right BalanceAlgorithm

Algorithm deleteRightBalance (ref root , ref shorter )Adjusts the balance factors, and balance the tree by rotating left if necessary

Pre tree is shorter

Post balance factors updated and balance restoredroot updatedshorter updated

1 if (root left-high)1 root > bal = EH

2 else if (root even-high)

1 root > bal = RH2 shorter = false

3 elseTree was right high already. Rotate left

Delete Right Balance

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Delete Right BalanceAlgorithm

3 elseTree was right high already. Rotate left

1 rightTree = root > right2 if (rightTree left-high)

Double rotation required1 leftTree = rightTree > left2 leftTree > bal = EH3 rotateRight (rightTree)4 rotateLeft (root)

3 elseSingle rotation required

Delete Right Balance

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gAlgorithm

Single rotation required1 if (rightTree even-high)

1 root > bal = RH2 rightTree > bal = LH3 shorter = false

2 else1 root > bal = EH2 rightTree > bal = EH

3 rotateLeft (root)4 returnEnd deleteRightBalance