binary tree (part 1)-chapter 6

7/29/2019 Binary Tree (Part 1)-Chapter 6

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Chapter 6

Binary Trees


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6.1 Trees, Binary Trees, and

Binary Search Trees

Linked lists usually are more flexible than arrays,

but it is difficult to use them to organize a

hierarchical representation of objects.

Stacks and queues reflect some hierarchy but arelimited to only one dimension.

For these reasons, create a new data type tree that

consists ofnodes and arcs/edges and has the root

at the top and the leaves (terminal nodes) at thebottom.

Tree Representation

Tree generally implemented in the computerusin ointers


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Tree Terminology / concepts

Path : a unique sequence of arcs or the resultingsequence of nodes.

Root: the node at the top of the tree. (one node /

one path)

Leaf : a node that has no children.

Subtree : any node can be considered to be the

root of a subtree, which consists of its children,

and its children's children, and so on. Level of the node :

The length of the path from the root to the node.

Level of a node is its distance from the root.


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Parent: Any node (except the root) has exactly oneedge running upward to another node..

It has a successor node

Child: Any node can have one or more lines

running downward to other nodes. It has a predecessor node

Siblings :Two or more nodes with the same

parentAncestor: Any node in the path from the root to the

node

Descendent :Any node in the path below the parent nodeAll nodes in the aths from a iven node to a leaf node

Tree Terminology cont


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Tree Terminology cont

Length : the number of arcs in a path. Height of non empty tree : the maximum level of a

node in the tree. Is the level of the leaf in the longest path from the root +

1 The height of an empty tree is -1

Depth == Height

Visiting : a node is visited when program control

arrives at the node, usually for the purpose ofcarrying out some operation on the node.

Traversing : to visit all the nodes in some specified

order.

Key : a value that is used to search for the item or


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8

A

B E F

C D G IH

Subtree B Root of Subtree I


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Binary Tree

It is a tree whose nodes have two children(possibly empty), and each child is designed as

either a left child or a right child.

If it has 2i nodes at each level i+1, it called

complete binary tree: all nonterminal nodes

have both their children and all leaves are at

the same level.

The binary search tree :Also called ordered binary trees

For each node n, all values stored in its left

subtree are less than value vstored in n, and

all values stored in the right subtree are greater


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xamples of binary trees


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Why tree / binary tree ?An ordered array is quick in search using

binary search , but it is slow in insertion

and deleting nodesLinked list is quick in insertion and

deletion but slow in searching

So, you may use a tree because itcombines the advantages of the other

two structures: an ordered array and a

linked list.

6.2 Implementing Binary Tree


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Binary tree can be implemented as :

1. A linked list.

2. An Array.

Declare a node as a structure with an informationfield and two pointers fields.

However, it may has problems when deleting and

inserting nodes


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. earc ng a nary earcTree

For every node, compare the key to be locatedwith 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.


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The complexity of searching is measured by the

number of comparisons performed.This number depends on the number of nodes

encountered on the unique path leading from

the root to the node being searched for.

So, the complexity is the length of the path

leading to this node plus 1.

Complexity depends on

1. The shape of the tree.2. The position of the node in the tree.


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Unbalanced Trees

Some of the trees are unbalanced, that is, theyhave most of their nodes on one side of the root

or the other. Individual subtrees may also be

unbalanced.


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1. Finding a Node

Finding a node with a specific key is the simplestof 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.


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Efficiency of the Find Operation

How long it takes to find a node depends on

how many levels down it is situated.

For example, there can be up to 31 nodes, but

no more than 5 levels. Thus you can find anynode using a maximum of only 5

comparisons.

This is O(log N) time.


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2. Inserting a Node

1. Find the place to insert it. This is much the sameprocess 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 greaterthan that of the parent.


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3. Deleting a Node

Deleting a node is the most complicatedcommon 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.


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Case1 : The node to be deleted has no

children

To delete a leaf node, change the appropriatechild field in the node's parent to point to null,

instead of to the node.


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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.


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Case 3 : The node to be deleted has two

children

If the deleted node has two children, you can'tjust replace it with one of these children, at least

if the child has its own children.

To delete a node with two children, replace thenode with its in order successor.


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4 Finding Maximum and Minimum


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4. Finding Maximum and Minimum

values

For the minimum, go to the left child of the root;then go to the left child of that child, and so on,

until you come to a node that has no left child.

This node is the minimum:


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For the maximum value in the tree, follow the

same procedure, but go from right child to

right child until you find a node with no rightchild. This node is the maximum.

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Summary

Trees consist of nodes (circles) connected by

edges (lines). The root is the topmost node in a tree; it has no

parent.

In a binary tree, a node has at most two children.

In a binary search tree, all the nodes that are left

descendants of node A have key values less than

A; all the nodes that are As right descendants

have key values greater than (or equal to) A. Trees perform searches, insertions, and deletions

in O(log N) time.

Nodes represent the data-objects being stored in

the tree.

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An unbalanced tree is one whose root has many

more left descendents than right descendants, orvice versa.

Searching for a node involves comparing the value

to be found with the key value of a node, and goingto that node's left child if the key search value is

less, or to the nodes right child if the search value

is greater.

Insertion involves finding the place to insert thenew node, and then changing a child data member

in its new parent to refer to it.

Summary (cont)

binary tree (part 1)-chapter 6

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