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Ch. 13-1 Balanced Search Trees

• Search time in a tree depends on the tree’s height• Balanced binary search trees

– Guarantees O(log n)-time search, insertion, and deletion

– AVL tree, red-black tree• Balanced k-ary trees

– 2-3 tree, 2-3-4 tree, B-trees


Worst balanced

Best balanced


2-3 Tree

• A special case of B-trees• k-ary search tree

TT11

key1 key2 key3 … keyk-1

TT22TTkkTT33 TT44 …

TTii TTi+1i+1< key i <


• T is a 2-3 tree of height h ifT is empty (height 0)

orT is of the form

where TL , TM , and TR are all 2-3 trees, each of height h-1. Here only TR can be empty w/ only one key in the node r.

TTLL

rr

TTRRTTMM


• 2-3 tree (intuitive description)– Each internal node has either two or three

children– All leaves are at the same level


A 2-3 Tree of Height 3


A 2-Node A 3-Node

Nodes in a 2-3 Tree3


An Example 2-3 Tree


A Balanced Binary Search Tree

A 2-3 Tree w/ the Same Elements


Continued… Continued… After a Sequence of InsertionsAfter a Sequence of Insertions


Node Structure

public class Tree23Node {private Comparable smallItem; private Comparable largeItem;private Tree23Node leftChild;private Tree23Node midChild;private Tree23Node rightChild;

}


A Sequence of Example Insertions in 2-3 Trees

Starting 2-3 Tree

40


inserting 39

40


inserting 38


inserting 37

38


inserting 36


inserting 35, 34, and 33


Inserting 32

32

30

50

35

34

33

37


Splitting a Leaf in a 2-3 TreeSplitting a Leaf in a 2-3 Tree


Splitting an Internal Node in a 2-3 TreeSplitting an Internal Node in a 2-3 Tree


Splitting the Root of a 2-3 TreeSplitting the Root of a 2-3 Tree


Insertion Algorithm

insertItem(theTree, key){

Find the leaf leafNode to which key should belong;Add key to leafNode;if (leafNode now has three items) { // overflow

split(leafNode);}

}


split (n) // n: node with 3 keys. If n is internal, it has 4 children.{

if (n is the root) {Create a new node p;

} else {Let p be the parent of n;

}Move up the middle key of n to p;Split the remaining two keys of node n into two nodes;Adjust links appropriately;if (p now has three items) {

split(p);}

}


Deletion in 2-3 Trees

• A sequence of example deletions

Starting 2-3 tree

40


Deleting 70

Binary search tree 에서의 삭제와 연관성을 생각해 볼 것


Deleting 100


DDeleting 80


b) Merging a leaf

a) Redistributing values


c) Redistributing values and children

d) Merging internal nodes


e) Deleting the root


Deletion AlgorithmdeleteItem(searchKey){

Find searchKey in the tree; if (searchKey is present) {

if (searchKey is not in a leaf) { Swap searchKey with its inorder successor, which should be in a leaf node, say theLeaf;

}Delete searchKey from theLeaf;if (theLeaf now has no item) {

fix(theLeaf);}return true;

} else {return false;

}}


fix (n) // n: empty node{

if (n is the root) {Remove the root;

} else {Let p be the parent of n;if (some sibling of n has more than one item) { // moving a spare

Distribute items appropriately among n, p, and the sibling;

if (n is internal) // adjust linkMove the appropriate child from sibling to n;

} else { // merge (no sibling has a spare, so mergible)Choose an adjacent sibling s of n;Bring the appropriate item down from p into s;if (n is internal) // at this point, n has only one child

Move n’s child to be s’s child;Remove node n;if (p is now empty) fix(p);

}}

}


2-3-4 Tree

• A special case of B-trees• k-ary search tree

TT11

key1 key2 key3 … keyk-1

TT22TTkkTT33 TT44 …

TTii TTi+1i+1< key i <


• T is a 2-3-4 tree of height h ifT is empty (height 0)

orT is of the form

where T1 , T2 , T3 , and T4 are all 2-3-4 trees, each of height h-1. Here T3 and T4 can be empty if there are fewer than three keys in the node r.

TT11

rr

TT33TT22 TT44


• 2-3-4 tree (intuitive description)– Each internal node has either two, three, or four

children– All leaves are at the same level


A 2-3-4 Tree of Height 3


A 4-Node in a 2-3-4 TreeA 4-Node in a 2-3-4 Tree


Node Structure

public class Tree234Node {private Comparable smallItem; private Comparable middleItem; private Comparable largeItem;private Tree234Node firstChild; private Tree234Node secondChild;private Tree234Node thirdChild;private Tree234Node fourthChild;

}


Insertion in 2-3-4 Trees• 4-node 를 만나자마자 무조건 split 한다 (2-3 tree 와 가장 다른 점 )• 그러므로 split 시점에 parent node 는 항상 key 를 더 받아들일

여유를 갖고 있다• 2-3 tree 의 insertion 보다 간명하다

InserInserting 20

• A sequence of example insertions

a node w/ 4 links

일단 split

20 삽입


Inserting 50 and 40


Inserting 70

일단 split 70 삽입


Inserting 80 and 15


Inserting 90

일단 split90 삽입


Inserting 100

일단 split

100 삽입


Splitting a 4-Node Root During InsertionSplitting a 4-Node Root During Insertion


Splitting a 4-Node Whose Parent is a 2-Node Splitting a 4-Node Whose Parent is a 2-Node During InsertionDuring Insertion


Splitting a 4-Node Whose Parent is a 3-NodeSplitting a 4-Node Whose Parent is a 3-NodeDuring InsertionDuring Insertion


Insertion Algorithm

insertItem(theTree, key){

Find the leaf leafNode to which key should belong;(If it encounters a 4-node on the path, split the node.)Add key to leafNode;

}

이 시점에 leafNode 에는 항상 key 를 담을 여유가 있다


Deletion in 2-3-4 Trees• Basically similar to that of 2-3 trees

– Find the node that contains the item to be deleted– If the node is internal, swap the item with its inorder successor (in

a leaf)

• If the leaf is a 3-node or a 4-node, simply delete the item

• To make the deletion simple, transform each 2-node (except the root) that you encounter during the search for the item into either a 3-node or a 4-node


Transformation of 2-Nodes (Sketch)

M


Red-Black Tree

• An equivalent binary-tree representation of 2-3-4 trees

• Each reference is given a color red or black– Assign the color black to the edges from the original 2-

3-4 tree – Assign the color red to the edges generated in the

course of transforming into a binary tree


Node Structure

public class RBTreeNode {public static final int RED = 0;public static final int BLACK = 1;private Comparable item; private RBTreeNode leftChild; private RBTreeNode rightChild;private int leftColor;private int rightColor;…

}


Insertion in Red-Black Trees

• 2-3-4 tree 의 insertion 과 기본적으로 같다 • 단지 binary tree 로 2-3-4 tree 의 insertion 을

simulation 한다


InserInserting 20

2-3-4 Tree Insertions and Red-Black Equivalents

10 60

30 10 60

30

20


Inserting 50 and 40

10 50

30

20 40 60

2-3-4 Tree Red-Black Tree


15 50

30

20 40 7010

60 80

2-3-4 Tree Red-Black Tree


Red-Black Representation of a 4-NodRed-Black Representation of a 4-Nod


Red-Black Representation of a 3-NodeRed-Black Representation of a 3-Node


2-3-4 Tree

Red-Black Tree


Splitting a Red-Black Representation of a 4-Node Splitting a Red-Black Representation of a 4-Node that is the Rootthat is the Root


Splitting a Red-Black Representation of a 4-Node Splitting a Red-Black Representation of a 4-Node whose Parent is a 2-Nodewhose Parent is a 2-Node


Splitting a Red-Black Representation of a 4-Node Splitting a Red-Black Representation of a 4-Node whose Parent is a 3-Nodewhose Parent is a 3-Node


Observation

• In a 2-3-4 tree– Every path from the root to a leaf encounters the same

# of edges• In a red-black tree

– Every path from the root to a leaf encounters the same # of black edges

– The # of red edges encountered in a path from the root to a leaf is not greater than “the # of black edges + 1”

– So, the longest path is not greater than “2*the shortest path + 1” (approx. two times the shortest path)


AVL Tree

• Devised by Adelson-Velskii and Landis • A balanced search tree

such that the heights of the left and right subtrees of any node differ by at most 1


Node Structure

public class AVLTreeNode {private Comparable item; private AVLTreeNode leftChild; private AVLTreeNode rightChild;private int leftHeight;private int rightHeight;…

}


AVL AVL

An unbalanced binary search tree

Balanced tree after rotation

Balanced tree after insertion


AVL

2

An Unbalanced Binary Search Tree

Balanced after a Single Left Rotation


Before and after a Single Left Rotation that Decreases the Tree’s Height

Height decreased!

2 AVL

not AVL


Height not changed!

2

not AVL AVL


An Unbalanced Binary Search Tree

Balanced after a Single Left Rotation


a) Before; b) During; and c) After a Double Rotation

2 35

30

22

25

10

20

3022

2510

20

35

(b) (c)

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