CS302 – Data Structures - Autonomous Robots Lab...Traversing 2-3 Trees •To traverse a 2-3 Tree...
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CS302 - Data Structures using C++ Kostas Alexis Topic: Balanced Search Trees - Introduction
Transcript of CS302 – Data Structures - Autonomous Robots Lab...Traversing 2-3 Trees •To traverse a 2-3 Tree...
CS302 - Data Structuresusing C++
Kostas Alexis
Topic: Balanced Search Trees - Introduction
2-3 Trees
• General Search Tree
• Interior nodes are either 2-nodes or 3-nodes
• 2-node has one data item and two children
• 3-node has two data items and three children• Simple implementations may use 3-nodes for both
2-3 Trees
• General Search Tree
• Interior nodes are either 2-nodes or 3-nodes
• 2-node has one data item and two children
• 3-node has two data items and three children• Simple implementations may use 3-nodes for both
50
leftChild rightChild
data
60|90
leftChild rightChildmiddleChild
small|large data
2-3 Trees
• General Search Tree
• Interior nodes are either 2-nodes or 3-nodes
• 2-node has one data item and two children
• 3-node has two data items and three children• Simple implementations may use 3-nodes for both
50
leftChild rightChild
data
60|90
leftChild rightChildmiddleChild
<50 >50
<60 >90>60,<90
small|large data
2-3 Trees
• General Search Tree
• Interior nodes are either 2-nodes or 3-nodes
• 2-node has one data item and two children
• 3-node has two data items and three children• Simple implementations may use 3-nodes for both
• Are never taller than minimum-height binary tree
• A 2-3 tree with n nodes never has height greater than log2(𝑛 + 1)
50
leftChild rightChild
data
60|90
leftChild rightChild
small|large data
middleChild
<50 >50
<60 >90>60,<90
2-3 Trees
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30 90
10 50
20 40
80 95
70
A Balanced Binary Search Tree
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30 70 | 90
40 60 9580
2-3 Tree with the same elements
10 | 20
2-3 Trees
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30 90
10 50
20 40
80 95
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A Balanced Binary Search Tree – Height = 4
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30 70 | 90
40 60 9580
2-3 Tree with the same elements – Height = 3
10 | 20
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40 50
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40 50 60
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40 50 60 70
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40 50 60 70
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40 50 60 70 80
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40 50 60 70 80
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40 50 60 70 80 90
Traversing 2-3 Trees
• To traverse a 2-3 Tree• Perform the analogue of an in-order traversal
• Leftmost subtree,
• Left value,
• Center subtree,
• Right value,
• Rightmost subtree
• Searching a 2-3 tree is as efficient assearching the shorted binary tree (withthe same values)
• 𝑂(log2𝑛)
50
30 70 | 90
40 60 958010 | 20
10 20 30 40 50 60 70 80 90 95
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
• Only exception – root (first value)• Create a 2-node and insert value
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
• Only exception – root (first value)• Create a 2-node and insert value
• Inserting a value into a 2-node turns it into a 3-node
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
• Only exception – root (first value)• Create a 2-node and insert value
• Inserting a value into a 2-node turns it into a 3-node
• Inserting a value into a 3-node causes it todivide
• Result is a subtree of 2-nodes
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
• Only exception – root (first value)• Create a 2-node and insert value
• Inserting a value into a 2-node turns it into a 3-node
• Inserting a value into a 3-node causes it todivide
• Result is a subtree of 2-nodes
50 70 90
50
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
• Only exception – root (first value)• Create a 2-node and insert value
• Inserting a value into a 2-node turns it into a 3-node
• Inserting a value into a 3-node causes it todivide
• Result is a subtree of 2-nodes
50 70 90
50 | 70
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
• Only exception – root (first value)• Create a 2-node and insert value
• Inserting a value into a 2-node turns it into a 3-node
• Inserting a value into a 3-node causes it todivide
• Result is a subtree of 2-nodes
50 70 90
50 | 70 | 90
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
• Only exception – root (first value)• Create a 2-node and insert value
• Inserting a value into a 2-node turns it into a 3-node
• Inserting a value into a 3-node causes it todivide
• Result is a subtree of 2-nodes
50 70 90
70
50 90
Adding to 2-3 Trees
• Inserting values into a 2-3 Tree• Always insert values into an EXISTING leaf
• Only exception – root (first value)• Create a 2-node and insert value
• Inserting a value into a 2-node turns it into a 3-node
• Inserting a value into a 3-node causes it todivide
• Result is a subtree of 2-nodes
50 70 90
70
50 90
• To insert an item with key k into a 2-3 tree• Locate the leaf at which the search for k would terminate• Insert the new item k into the leaf
• If the leaf now contains only two items – done
• If the leaf now contains three items, split the leaf into two nodes and move the
middle value into parent
Adding to 2-3 Trees
• When an internal node contains three items• Split the node into two nodes
• Accommodate the node’s children
• When the root contains three items• Split the root into two nodes
• Create a new root node
• Tree grows in height
Adding to 2-3 Trees
• When an internal node contains three items• Split the node into two nodes
• Accommodate the node’s children
• When the root contains three items• Split the root into two nodes
• Create a new root node
• Tree grows in height
Adding to 2-3 Trees
• When an internal node contains three items• Split the node into two nodes
• Accommodate the node’s children
• When the root contains three items• Split the root into two nodes
• Create a new root node
• Tree grows in height
20
50
30
80
25
60
50
Order of Values Given
20 90 30 80 25 60 50
Adding to 2-3 Trees
• When an internal node contains three items• Split the node into two nodes
• Accommodate the node’s children
• When the root contains three items• Split the root into two nodes
• Create a new root node
• Tree grows in height
20
50
30
80
25
60
50
Order of Values Given
20 90 30 80 25 60 50
20
Adding to 2-3 Trees
• When an internal node contains three items• Split the node into two nodes
• Accommodate the node’s children
• When the root contains three items• Split the root into two nodes
• Create a new root node
• Tree grows in height
20
50
30
80
25
60
50
Order of Values Given
20 90 30 80 25 60 50
20 | 90
Adding to 2-3 Trees
• When an internal node contains three items• Split the node into two nodes
• Accommodate the node’s children
• When the root contains three items• Split the root into two nodes
• Create a new root node
• Tree grows in height
20
50
30
80
25
60
50
Order of Values Given
20 90 30 80 25 60 50
30
20 90
Adding to 2-3 Trees
• When an internal node contains three items• Split the node into two nodes
• Accommodate the node’s children
• When the root contains three items• Split the root into two nodes
• Create a new root node
• Tree grows in height
20
50
30
80
25
60
50
Order of Values Given
20 90 30 80 25 60 50
30 | 80
20 | 25 50 | 60 90
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
30 70 | 90
10 | 20 40 60 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
30 70 | 90
10 | 20 40 60 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
30 70 | 90
10 40 60 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
30 70 | 90
10 40 60 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
30 70 | 90
10 40 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
30 90
10 40 70 | 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
30 90
10 40 70 | 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
40 90
10 30 70 | 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50
90
10 | 40 70 | 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50 90
10 | 40 70 | 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50 | 90
10 | 40 70 | 80 95
Removing From a 2-3 Tree
• Removing values from 2-3 trees• Always remove from a leaf
• Values (and children) will be redistributed
• Nodes can be merged
• Only root node is deleted• And only if it is empty (contains no values)
50 | 90
10 | 40 70 | 80 95
Thank you
