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2-3 Trees: Data Structures and Algorithms, Slides of Data Structures and Algorithms

Dhirubhai Ambani Institute of Information and Communication Technology Data Structures and Algorithms

An in-depth exploration of 2-3 trees, a self-balancing tree data structure. It covers the basics of 2-3 trees, their advantages and disadvantages, insertion and deletion algorithms, and node splitting. The document also includes examples and observations.

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CPSC 223

Algorithms & Data Abstract Structures

Lecture 22: !

B Trees: 2-3 & 2-3-4 trees!

Today …

• 2-3 Trees [Ch 12: 651-670]!

• 2-3-4 Trees (insertion) !

CPSC%223%++%Fall%2010%

Discover Slides of Data Structures and Algorithms Dhirubhai Ambani Institute of Information and Communication Technology

Partial preview of the text

Download 2-3 Trees: Data Structures and Algorithms and more Slides Data Structures and Algorithms in PDF only on Docsity!

CPSC 223

Algorithms & Data Abstract Structures

Lecture 22:

B Trees: 2-3 & 2-3-4 trees

Today …

2-3 Trees [Ch 12: 651-670]
2-3-4 Trees (insertion) CPSC 223 -‐-‐ Fall 2010

2-3 Trees

CPSC 223 -‐-‐ Fall 2010

B Trees (… starting with 2-3 trees)

A “ 2-3 Tree ” is a (search) tree in which:
- Every internal (non-leaf) node has either 2 or 3 children
- All leaves are at the same level
A node with 2 children is called a “ 2-node ”
A node with 3 children is called a “ 3-node ” CPSC 223 -‐-‐ Fall 2010

Placing Data in a 2-3 Tree

A 2-node (2 children) must contain a single data item

The item’s search key S is:

greater than its left child’s search keys
smaller than (or equal to) its middle child’s search keys

A 3-node (3 children) must contain two data items

For S the left and L the right search key:

S is greater than the left child’s search keys
S is smaller than (or equal to) the middle child’s search keys
L is greater than the middle child’s search keys
L is smaller than (or equal to) the right child’s search keys

A leaf node may contain one or two data items CPSC 223 -‐-‐ Fall 2010 S L S

Placing Data in a 2-3 Tree

Is this a well-formed 2-3 tree?
- NO! CPSC 223 -‐-‐ Fall 2010 6 11 4 5 10

2-‐node with 2 items!

Traversing a 2-3 Tree

• Traversal very similar to a BST!

void inorder (const Node* root) const

if(root != NULL) {

inorder (root->leftChild);

visit (root->item1); // visit 1st^ data item

inorder (root->middleChild);

if( has2Keys (root)) {

visit(root->item2); // visit 2nd^ data item

inorder (root->rightChild);

CPSC 223 -‐-‐ Fall 2010

Traversing a 2-3 Tree (inorder)

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root)

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) New root Traverse left Traverse left

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) New root New root Traverse left Traverse left

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) New root New root (^) Visit first item (after traversing to NULL node) Output: 10 Traverse left Traverse left

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) New root New root (^) Visit second item (after traversing to NULL node) Output: 10 20 Traverse left Traverse left

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) New root Traverse middle Output: 10 20 30 Traverse left New root

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) New root Traverse middle Output: 10 20 30 40 Traverse left Visit first item (after traversing to NULL node) New root

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) New root Output: 10 20 30 40 Traverse left Finished visiting nodes

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) Output: 10 20 30 40 50 Visit first item (after traversing to NULL node)

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) Output: 10 20 30 40 50 Traverse middle New root Traverse left

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) Output: 10 20 30 40 50 Traverse middle New root Traverse left New root

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) Output: 10 20 30 40 50 60 Traverse middle New root Traverse left New root Visit first item (after traversing to NULL node)

Traversing a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) Output: 10 20 30 40 50 60 65 Traverse middle New root Visit first item (after traversing to NULL node)

Searching a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) Lookup “70”

Searching a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 Start here (root) Lookup “70”

Since 2-node and 50 < 70

Search middle child

Searching a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 New root Lookup “70”

Searching a 2-3 Tree

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90 New root Lookup “70”

Since 3-node and 65 < 70 < 80

Search middle child

2-3 Trees

Some Observations

A 2-3 tree is (obviously) not a binary tree
- But a 2-3 tree is very similar to a full binary tree
- All leaves are on the same level in a 2-3 tree
So a 2-3 tree is always (perfectly) balanced
- … and has height at most ⎡log 2 ( n +1)⎤
- which is the case for a full BST
- Best case height is ⎡log 3 ( n +1)⎤ (each node has 3 children) CPSC 223 -‐-‐ Fall 2010

2-3 Trees

Advantages/Disadvantages

2-3 trees are (fully) balanced … O (log n ) height
- Insertion and deletion not (terribly) difficult to implement
Shorter height (because of “wider” nodes) does not

necessarily mean more efficient retrieval time

Requires additional comparisons (up to 2 per node)
Height helps in some cases (e.g., File Indexes/Databases) CPSC 223 -‐-‐ Fall 2010

Inserting Nodes

So how can we maintain a 2-3 tree on insertion?

Locate the leaf node where the item “should” be
If the leaf contains only one item, then we are done!
Otherwise, we have to “ split ” the node
Splitting a node may cause the parent to split
And so on up the tree CPSC 223 -‐-‐ Fall 2010

Insertion into a 2-3 Tree

1. Search for item

CPSC 223 -‐-‐ Fall 2010 50 30 65 80 10 20 40 60 70 90

2-3 Trees: Data Structures and Algorithms, Slides of Data Structures and Algorithms

Related documents

Partial preview of the text

Download 2-3 Trees: Data Structures and Algorithms and more Slides Data Structures and Algorithms in PDF only on Docsity!

CPSC 223

Algorithms & Data Abstract Structures

Lecture 22:

B Trees: 2-3 & 2-3-4 trees

Today …

2-3 Trees

B Trees (… starting with 2-3 trees)

Placing Data in a 2-3 Tree

The item’s search key S is:

For S the left and L the right search key:

Placing Data in a 2-3 Tree

2-­‐node with 2 items!

Traversing a 2-3 Tree

• Traversal very similar to a BST!

void inorder (const Node* root) const

if(root != NULL) {

inorder (root->leftChild);

visit (root->item1); // visit 1st^ data item

inorder (root->middleChild);

if( has2Keys (root)) {

visit(root->item2); // visit 2nd^ data item

inorder (root->rightChild);

Traversing a 2-3 Tree (inorder)

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Traversing a 2-3 Tree

Searching a 2-3 Tree

Searching a 2-3 Tree

Since 2-node and 50 < 70

Search middle child

Searching a 2-3 Tree

Searching a 2-3 Tree

Since 3-node and 65 < 70 < 80

Search middle child

2-3 Trees

Some Observations

2-3 Trees

Advantages/Disadvantages

necessarily mean more efficient retrieval time

Inserting Nodes

So how can we maintain a 2-3 tree on insertion?

Insertion into a 2-3 Tree

1. Search for item

Insert “75”

2-‐node with 2 items!