tree.ppt

TREE
 1. THE CONCEPT OF TREES
 2. BINARY TREE AND REPRESENTATION
 3. BINARY TREE TRAVERSAL
 4. BINARY SEARCH TREE
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1. THE CONCEPT OF TREES
 A tree is a set of one or more nodes T:
– there is a specially designated node called a
root
– The remaining nodes are partitioned into n
disjointed set of nodes T1, T2,…,Tn, each of
which is a tree.

 Example

 It’s not a tree
Tree

1. THE CONCEPT OF TREES: Some
terminology
 Root
 Child (left,right)
 Parent
 Leaf node
 Subtree
 Ancestor of a node
 Descendant of a node

 Degree of a Node of a Tree
– The degree of a node of a tree is the number of subtrees
having this node as a root.
 Degree of a Tree
– The degree of a tree is defined as the maximum of degree
of the nodes of the tree
 Level of a Node
– level of the root node as 1, and incrementing it by 1 as we
move from the root towards the subtrees.

2. BINARY TREE AND REPRESENTATION
 BINARY TREE
– no node can have a degree of more than 2.
– The maximum number of nodes at level i will be
2i−1
– If k is the depth of the tree then the maximum
number of nodes that the tree can have is
– 2k − 1 = 2k−1 + 2k−2 + … + 20

 BINARY TREE
– A full binary tree is a binary of depth k having
2k − 1 nodes.
– If it has < 2k − 1, it is not a full binary tree

What is the height h of a full tree
with N nodes?
 The max height of a tree with N nodes is N
(same as a linked list)
 The min height of a tree with N nodes is
log(N+1)
2 1
2 1
log( 1) (log )
h
h
N
N
h N O N
 =
 = 
 =  

3=22-1
7=23-1 15=24-1
full binary

struct node
{ int data;
node *left;
node *right;
};

Tree traversal
 Used to print out the data in a tree in a certain order
– inorder (LDR )
– Postorder (LRD )
– preorder (DLR )
 Pre-order traversal
– Print the data at the root
– Recursively print out all data in the left subtree
– Recursively print out all data in the right subtree

Preorder, Postorder and Inorder
 Preorder traversal
– node, left, right
– prefix expression
++a*bc*+*defg

 Postorder traversal
– left, right, node
– postfix expression
abc*+de*f+g*+
 Inorder traversal
– left, node, right.
– infix expression
a+b*c+d*e+f*g

3. BINARY TREE TRAVERSAL
Inorder = DBEAC
Many trees

4. BINARY SEARCH TREE
 A binary search tree
– is a binary tree (may be empty)
– every node must contain an identifier.
– An identifier of any node in the left subtree is less
than the identifier of the root.
– An identifier of any node in the right subtree is
greater than the identifier of the root.
– Both the left subtree and right subtree are binary
search trees.

 binary search tree.

Binary Search Trees
A binary search tree Not a binary search tree

Performance
 Consider a dictionary with n
items implemented by
means of a binary search
tree of height h
– the space used is O(n)
– methods find, insert and
remove take O(h) time
 The height h is O(n) in the
worst case and O(log n) in
the best case

 Why using binary search tree
– traverse in inorder: sorted list
– searching becomes faster
 But..
– Insert, delete: slow
 Important thing: Index in Database system
– Using the right way of Index property

Search
 To search for a key k, we
trace a downward path
starting at the root
 The next node visited
depends on the outcome
of the comparison of k with
the key of the current node
 If we reach a leaf, the key
is not found and we return
nukk
 Example: find(4):
– Call TreeSearch(4,root)
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Insertion
 To perform operation inser(k,
o), we search for key k (using
TreeSearch)
 Assume k is not already in the
tree, and let let w be the leaf
reached by the search
 We insert k at node w and
expand w into an internal node
 Example: insert 5
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w
w

 Insert new node

 Insert new node
void InsertNode(node* &root,node *newnode)
{
if (root==NULL)
root=newnode;
else
if (root->data>newnode->data)
InsertNode(root->l,newnode);
else
if (root->data<newnode->data)
InsertNode(root->r,newnode);
}

 traverse node
void preorder(node* r)
{
if (r!=NULL)
{ cout<<r->data<<" ";
inorder(r->l);
inorder(r->r);
}
}

 traverse node

tree.ppt

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tree.ppt