meljun cortes data structures types of binary search tree

8/21/2019 Meljun Cortes Data Structures Types of Binary Search Tree

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Data Structures

Balanced / unbalanced tree

Balanced tree - equal items on each subtree ofnode to minimize the maximum path from the

root to any leaf node

Height-balanced - left and right subtree

height of each node is within 1

Types of Binary Search Tree *Property of STI Page 1 of 24

Unbalanced tree - root has more left

descendants than its right descendants or vice-versa

Self-balancing tree - attempt to keep its height,the number of levels of nodes underneath theroot, as small as possible

Tree rotation - changes the structure of the treewithout intervening any element in the tree

especially its order


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Data Structures

AVL Tree

named after G.M. Adelso-Velsky and E.M.Landis

can also be called as height balanced

Balance factor - done by right subtree less the

height of its left subtree Insertion - as it is inserting in an unbalanced

BST and retraces the one’s steps toward theroot

–


Search - no provisions will be taken and will notchange the structure of the tree

an AVL tree because itshows that each leftsubtree has a height of 1

greater than each rightsubtree

not an AVL tree for thesubtree of E has theheight of 4 and thesubtree of I has theheight of 2


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Data Structures

Red-Black Tree

a special type of BST that organizes pieces ofcomparable data such as numbers

two characteristics that this tree comprises

The nodes are colored

During insertion and deletion, rules arefollowed that preserve variousarrangements of these colors

Red-Black Rules


The root is always black.

If a node is red, its children must beblack.

Both children of every red node areblack.

Every path from the root to a leaf, or toa null child, must contain the samenumber of black nodes.

things should be done when tree is notbalanced or the color rules are violated

Change the colors of the nodes.

Perform rotations.


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Data Structures

Red-Black Tree

Rotations are designed to:

Raise some nodes and lower others to helpbalance the tree

Ensure that the characteristics of a binary

search tree are not violated To maintain the properties of red-black trees

when inserting a node

Enter a new element on a tree as a red leaf


Maintain the root as a black node

When moving down the path to look onwhere to insert, update the colors of thenodes when you find a black parent with twored children

Inserting a node at the bottom may result intwo successive red nodes; employ rotationto reorder the coloring of the tree


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Data Structures

Splay Trees

elements in the tree that have been accessedrecently can be accessed very quickly

invented by Daniel Sleator and Robert Tarjan

it performs an operation called “splaying” also

know as rotation

Advantages

Faster performance


Easier implementation

Efficient with identical keys


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Data Structures

Splay Trees

Three splay steps:

Zig step

Zig-zig step


Zig-zag step


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Data Structures

2-3-4 Trees

Multiway tree - allowing having more data itemsand children per node


2-3-4 tree – a multiway tree that can have up to4 children and 3 data items per node

Possible arrangement for non-leaf nodes A node with one data item always has two

children

A node with two data items always has

three children

A node with three data items always hasfour children


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Data Structures

2-3-4 Trees



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Data Structures

2-3-4 Trees

Principle

All children in the subtree rooted at child 0have key values less than key 0

All children in the subtree rooted at child 1

have key values greater than 0 but less thankey 2

All children in the subtree rooted at child 2have key values greater than key 1 but lessthan ke 2


All children in the subtree rooted at child 3have key values greater than key 2


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Data Structures

Find key value 64

2-3-4 Trees



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Data Structures

Insert operations

2-3-4 Trees



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Data Structures

Splitting a node

A new, empty node is created. It’s a siblingof the node being split, and is placed to its

right.

Data item C is moved into a new node. Data item B is moved into the parent of the

node being split.

Data item A remains where it is.

2-3-4 Trees


from the node being split and connected tothe new node


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Data Structures

2-3-4 Trees


Split a root A new node is created that becomes the new

root and the parent of the node being split.

A second new node is created that becomes asibling of the node being split.

Data item C is moved into the new sibling.

Data item B is moved into the new root.

Data item A remains where it is.

The two rightmost children of the node beingsplit are disconnected from it and connected tothe new right—hand node.


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Data Structures

2-3-4 trees



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Data Structures

“B” – balanced, Boieng, Bayer

keeps data in a sorted order

Database and filesystems – applications thatuses this data structure

require that all leaf nodes are at the same depth Search, insert, delete are operations that can be

performed

overflowed - adding on a full leaf node

B-trees


To make nodes in order

Half the leaf node making it the left child

Pick the middle and place it on the root Create a new node placing the remaining

values


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Data Structures

B-trees

Insert 18

simply move the values of the middle leaf nodeto insert 18


Insert 46


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Data Structures

From the tree below, insert the following values:

67

54

39

Exercise



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Data Structures

Inserting 67

Inserting 54

Answers


Inserting 39


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Data Structures

a partial sorted binary tree

Heap

Complete

Not complete


Implemented as array

satisfies the heap condition


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Data Structures

Removal of node in heap means to remove themaximum key which this node is always the root

Remove the root

Move the last node into the root

Trickle (bubble) the last node down until it’sbelow a larger node and above a smallerone

Heap



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Data Structures

Heap



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Data Structures

Heap



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Data Structures

Heap

Insertion – use trickle up



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Data Structures

Heap

Types of Binary Search Tree *Property of STI

meljun cortes data structures types of binary search tree

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