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1

Binary Search Trees

Nelson Padua-Perez

Chau-Wen Tseng

Department of Computer Science

University of Maryland, College Park

Overview

Binary trees

Binary search trees

Find

Insert

Delete

2

Hierarchical Data Structures

One-to-many relationship between elements

Tree

Single parent, multiple children

Binary tree

Tree with 0-2 children per node

Tree Binary Tree

Trees

Terminology

Root no predecessor

Leaf no successor

Interior non-leaf

Height distance from root to leaf

Root node

Leaf nodes

Interior nodes

Height

3

Types of Binary Trees

Degenerate - only one child

Balanced - mostly two children

Complete - always two children

Degenerate

binary treeBalanced binary treeComplete binary tree

Binary Trees Properties

Degenerate

Height = O(n) for n

nodes

Similar to linear list

Balanced

Height = O( log(n) )

for n nodes

Useful for searches

Degenerate

binary treeBalanced binary tree 4

Binary Search Trees

Key property

Value at node

Smaller values in left subtree

Larger values in right subtree

Example

X >Y X Binary Search Trees

Examples

Binary

search treesNon-binary search tree5 10 30
22545
510
45
22530
5 10 302
2545
5

Binary Tree Implementation

Class Node {

Value data;

Node left, right; // null if empty

void insert( Value data1 ) { ... } void delete( Value data2 ) { ... }

Node find( Value data3 ) { ... }

Polymorphic Binary Tree Implement.

Interface Node {

Node insert( Value data1 ) { ... }

Class EmptyNode implements Node {

Node insert( Value data1 ) { ... }

Class NonEmptyNode implements Node {

Value data;

Node left, right; // Either Empty or NonEmpty

void insert( Value data1 ) { ... } 6

Iterative Search of Binary Tree

Node Find( Node n, Value 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;

Find( root, keyValue );

Recursive Search of Binary Tree

Node Find( Node n, Value 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);

Find( root, keyValue );

7

Example Binary Searches

Find ( 2 )

510
30

225 45

5 10 302
2545

10 > 2, left

5 > 2, left

2 = 2, found5 > 2, left

2 = 2, found

Example Binary Searches

Find ( 25 )

510
30
22545
5 10 302
2545

10 < 25, right

30 > 25, left

25 = 25, found5 < 25, right

45 > 25, left

30 > 25, left

10 < 25, right

25 = 25, found

8

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

Requires

Ability to comparekey values

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

9

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 )

510
30

2254510 < 20, right

30 > 20, left

25 > 20, left

Insert 20 on left

20 10

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

ORsmallest value Z on right subtree

b) Deletereplacement value (Y or Z) from subtree

Observation

O( log(n) ) operation for balanced tree

Deletions may unbalance tree

Example Deletion (Leaf)

Delete ( 25 )

510
30

2254510 < 25, right

30 > 25, left

25 = 25, delete510

30
245
11

Example Deletion (Internal Node)

Delete ( 10 )

510
30

2254555

30

2254525

30

225 45

Replacing 10

with largest value in left subtreeReplacing 5 with largest value in left subtreeDeleting leaf

Example Deletion (Internal Node)

Delete ( 10 )

510
30

22545525

30

22545525

30
245

Replacing 10

with smallest value in right subtreeDeleting leaf Resulting treequotesdbs_dbs19.pdfusesText_25