Chapter 7 Trees_Part31

SEARCH TREE

2

Search Trees

Two standard search trees:

Binary Search Trees (non-balanced)All items in left sub-tree are less than rootAll items in right sub-tree are greater than or

equal to the rootEach sub-tree is a binary search tree

AVL trees (balanced)

3

Binary Search Trees

All items in left sub-tree are less than root All items in right sub-tree are greater than or equal

to the root Each sub-tree is a binary search tree

Searching a Binary Tree

For every node, compare the key to be located with the value stored in the node currently pointed at.

1. If key is less than the value, go to the left subtree and try again.

2. If it is greater than that value, try the right subtree.

3. If it is the same, the search stops.

4. The search is aborted if there is no way to go – the key is not in the tree.

1. Finding a Node

Finding a node with a specific key is the simplest of the major tree operations. Remember that the nodes in a binary search tree correspond to objects containing information, one of them can be considered as a key.

6

Example

(a) A binary search tree T representing a dictionary D with integer keys; (b) nodes of T visited when executing operations find(76) (successful) and find(25) (unsuccessful) on D. For simplicity, we show keys but entry values.

7

2. Inserting a Node

1. Find the place to insert it. This is much the same process as trying to find a node which turns out not to exist.

2. Follow the path from the root to the appropriate node, which will be the parent of the new node.

3. After this parent is found, the new node is connected as its left or right child, depending on whether the new node's key is less or greater than that of the parent.

10

11

Example

Insertion of an entry with key 78 into the search tree . Finding the position to insert is shown in (a), and the resulting tree is shown in (b).

12

Insert

13

Insert

14

3. Deleting a Node

Deleting a node is the most complicated common operation required for binary search trees. However, deletion is important in many tree applications.

There are three cases to consider:-

1- The node to be deleted is a leaf node (has no children).

2- The node to be deleted has one child.

3- The node to be deleted has two children.

Case1 : The node to be deleted has no children

To delete a leaf node, change the appropriate child field in the node's parent to point to null, instead of to the node.

Case 2 : The node to be deleted has one child

The node has only two connections: to its parent and to its only child. You want to "snip" the node out of this sequence by connecting its parent directly to its child.

Case 3 : The node to be deleted has two children

the node has 2 children replace the key of that node with the minimum element at the right

subtree delete the minimum element

Has either no child or only right child because if it has a left child, that left child would be smaller and would have been chosen.

19

20

21

2- AVL Trees

AVL trees are height-balanced binary search trees Height-Balance Property: For every internal node

v of T, the heights of the children of v differ by at most 1.

22

2- AVL Trees• Height-balanced binary search tree where the heights of sub-

trees differ by no more than 1:

| HL – HR | <= 1• Search effort for an AVL tree is O(log2n)

• Each node has a balance factor• Balance factors may be:

– Left High (LH) = +1 (left sub-tree higher than right sub-tree)

– Even High (EH) = 0 (left and right sub-trees same height)– Right High (RH) = -1 (right sub-tree higher than left sub-

tree)

23 Figure 8-12

24 Figure 8-13 An AVL tree

25

Balancing Trees

Insertions and deletions potentially cause a tree to be imbalanced When a tree is detected as unbalanced, nodes are balanced by

rotating nodes to the left or right Four imbalance cases:

Left of left: the sub-tree of a left high tree has become left high

Right of right: the sub-tree of a right high tree has become right high

Right of left: the sub-tree of a left high tree has become right high

Left of right: the sub-tree of a right high tree has become left high

Each imbalance case has simple and complex rotation cases

26

27

28

Case 1: Left of Left

• The sub-tree of a left high tree has become left high

• Simple right rotation: Rotate the out of balance node (the root) to the right

• Complex right rotation: Rotate root to the right, so the old root is the right sub-tree of the new root; the new root's right sub-tree is connected to the old root's left sub-tree

29

30

Case 2: Right of Right• Mirror of Case 1: The sub-tree of a right

high tree has become right high• Simple left rotation: Rotate the out of

balance node (the root) to the left• Complex left rotation: Rotate root to the

left, so the old root is the left sub-tree of the new root; the new root's left sub-tree is connected to the old root's right sub-tree

31

32

Case 3: Right of Left

• The sub-tree of a left high tree has become right high

• Simple double rotation right: Rotate left sub-tree to the left; rotate root to the right, so the old root's left node is the new root

• Complex double rotation right: Rotate the right-high left sub-tree to the left; rotate the left-high left sub-tree to the right

33

34

Case 4: Left of Right

• Mirror of Case 3: The sub-tree of a right high tree has become left high

• Simple double rotation right: Rotate right sub-tree to the right; rotate root to the left, so the old root's right node is the new root

• Complex double rotation right: Rotate the left-high right sub-tree to the right; rotate the right-high right sub-tree to the left

35

36AVL Node Structure

Node

key <keyType>

data <dataType>

Left <pointer to Node>

right <pointer to Node>

bal <LH, EH, RH> // Balance factor

End Node

A tree was defined as either an empty structure or a structure whose children are disjoint trees.

This means that each node in some kind of trees can have more than two children. This tree is called a multiway tree of order m, or m-

way tree

Multiway Search Tree

38

Multiway Search Tree con..

A multiway search tree of order m is a multiway tree in which :

1. Each node has m children and m-1 keys.

2. The keys in each node are in ascending order.

3. The keys in the first i children are smaller than the ith key.

4. The keys in the last m-i children are larger than the ith key.

B- Trees operates closely with secondary storage. Property: the size of a node can be made as large as a

block. A B- tree of order m is a multi way search tree with the

following properties:

1. The root has at least two subtrees unless it is a leaf.

2. Each nonroot and each nonleaf node holds k-1 keys and k pointers to subtrees where [m/2] ≤ k ≤ m.

3. Each leaf node holds k-1 keys where [m/2] ≤ k ≤ m.

4. All leaves are usually on the same level.

B-Tree

B-Tree

42

References:• Text book, chapter 8: Binary Trees

End Of Chapter