1.5 binary search tree

Tree Data Structures
S. Sudarshan
Based partly on material from
Fawzi Emad & Chau-Wen Tseng

Trees Data Structures
 Tree
 Nodes
 Each node can have 0 or more children
 A node can have at most one parent
 Binary tree
 Tree with 0–2 children per node
Tree Binary Tree

Trees
 Terminology
 Root ⇒ no parent
 Leaf ⇒ no child
 Interior ⇒ non-leaf
 Height ⇒ distance from root to leaf
Root node
Leaf nodes
Interior nodes Height

Binary Search Trees
 Key property
 Value at node
 Smaller values in left subtree
 Larger values in right subtree
 Example
 X > Y
 X < Z
Y
X
Z

Binary Search Trees
 Examples
Binary
search trees
Not a binary
search tree
5
10
30
2 25 45
5
10
45
2 25 30
5
10
30
2
25
45

Binary Tree Implementation
Class Node {
int data; // Could be int, a class, etc
Node *left, *right; // null if empty
void insert ( int data ) { … }
void delete ( int data ) { … }
Node *find ( int data ) { … }
…
}

Iterative Search of Binary Tree
Node *Find( Node *n, int key) {
while (n != NULL) {
if (n->data == key) // Found it
return n;
if (n->data > key) // In left subtree
n = n->left;
else // In right subtree
n = n->right;
}
return null;
}
Node * n = Find( root, 5);

Recursive Search of Binary Tree
Node *Find( Node *n, int key) {
if (n == NULL) // Not found
return( n );
else if (n->data == key) // Found it
return( n );
else if (n->data > key) // In left subtree
return Find( n->left, key );
else // In right subtree
return Find( n->right, key );
}
Node * n = Find( root, 5);

Example Binary Searches
 Find ( root, 2 )
5
10
30
2 25 45
5
10
30
2
25
45
10 > 2, left
5 > 2, left
2 = 2, found
5 > 2, left
2 = 2, found
root

Example Binary Searches
 Find (root, 25 )
5
10
30
2 25 45
5
10
30
2
25
45
10 < 25, right
30 > 25, left
25 = 25, found
5 < 25, right
45 > 25, left
30 > 25, left
10 < 25, right
25 = 25, found

Types of Binary Trees
 Degenerate – only one child
 Complete – always two children
 Balanced – “mostly” two children
 more formal definitions exist, above are intuitive ideas
Degenerate
binary tree
Balanced
binary tree
Complete
binary tree

Binary Trees Properties
 Degenerate
 Height = O(n) for n
nodes
 Similar to linked list
 Balanced
 Height = O( log(n) )
for n nodes
 Useful for searches
Degenerate
binary tree
Balanced
binary tree

Binary Search Properties
 Time of search
 Proportional to height of tree
 Balanced binary tree
 O( log(n) ) time
 Degenerate tree
 O( n ) time
 Like searching linked list / unsorted array

Binary Search Tree Construction
 How to build & maintain binary trees?
 Insertion
 Deletion
 Maintain key property (invariant)
 Smaller values in left subtree
 Larger values in right subtree

Binary Search Tree – Insertion
 Algorithm
1. Perform search for value X
2. Search will end at node Y (if X not in tree)
3. If X < Y, insert new leaf X as new left subtree for
Y
4. If X > Y, insert new leaf X as new right subtree
for Y
 Observations
 O( log(n) ) operation for balanced tree
 Insertions may unbalance tree

Example Insertion
 Insert ( 20 )
5
10
30
2 25 45
10 < 20, right
30 > 20, left
25 > 20, left
Insert 20 on left
20

Binary Search Tree – Deletion
 Algorithm
1. Perform search for value X
2. If X is a leaf, delete X
3. Else // must delete internal node
a) Replace with largest value Y on left subtree
OR smallest value Z on right subtree
b) Delete replacement value (Y or Z) from subtree
Observation
 O( log(n) ) operation for balanced tree
 Deletions may unbalance tree

Example Deletion (Leaf)
 Delete ( 25 )
5
10
30
2 25 45
10 < 25, right
30 > 25, left
25 = 25, delete
5
10
30
2 45

Example Deletion (Internal Node)
 Delete ( 10 )
5
10
30
2 25 45
5
5
30
2 25 45
2
5
30
2 25 45
Replacing 10
with largest
value in left
subtree
Replacing 5
with largest
value in left
subtree
Deleting leaf

Example Deletion (Internal Node)
 Delete ( 10 )
5
10
30
2 25 45
5
25
30
2 25 45
5
25
30
2 45
Replacing 10
with smallest
value in right
subtree
Deleting leaf Resulting tree

Balanced Search Trees
 Kinds of balanced binary search trees
 height balanced vs. weight balanced
 “Tree rotations” used to maintain balance on insert/delete
 Non-binary search trees
 2/3 trees
 each internal node has 2 or 3 children
 all leaves at same depth (height balanced)
 B-trees
 Generalization of 2/3 trees
 Each internal node has between k/2 and k children
 Each node has an array of pointers to children
 Widely used in databases

Other (Non-Search) Trees
 Parse trees
 Convert from textual representation to tree
representation
 Textual program to tree
 Used extensively in compilers
 Tree representation of data
 E.g. HTML data can be represented as a tree
 called DOM (Document Object Model) tree
 XML
 Like HTML, but used to represent data
 Tree structured

Parse Trees
 Expressions, programs, etc can be
represented by tree structures
 E.g. Arithmetic Expression Tree
 A-(C/5 * 2) + (D*5 % 4)
+
- %
A * * 4
/ 2 D 5
C 5

Tree Traversal
 Goal: visit every node of a tree
 in-order traversal
void Node::inOrder () {
if (left != NULL) {
cout << “(“; left->inOrder(); cout << “)”;
}
cout << data << endl;
if (right != NULL) right->inOrder()
}Output: A – C / 5 * 2 + D * 5 % 4
To disambiguate: print brackets
+
- %
A * * 4
/ 2 D 5
C 5

Tree Traversal (contd.)
 pre-order and post-order:
void Node::preOrder () {
if (left != NULL) left->preOrder ();
if (right != NULL) right->preOrder ();
}
void Node::postOrder () {
if (left != NULL) left->preOrder ();
if (right != NULL) right->preOrder ();
}
Output: + - A * / C 5 2 % * D 5 4
Output: A C 5 / 2 * - D 5 * 4 % +
+
- %
A * * 4
/ 2 D 5
C 5

XML
 Data Representation
 E.g.
<dependency>
<object>sample1.o</object>
<depends>sample1.cpp</depends>
<depends>sample1.h</depends>
<rule>g++ -c sample1.cpp</rule>
</dependency>
 Tree representation
dependency
object depends
sample1.o sample1.cpp
depends
sample1.h
rule
g++ -c …

Graph Data Structures
 E.g: Airline networks, road networks, electrical circuits
 Nodes and Edges
 E.g. representation: class Node
 Stores name
 stores pointers to all adjacent nodes
 i,e. edge == pointer
 To store multiple pointers: use array or linked list
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1.5 binary search tree

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