Binary Search tree, redblack tree, and AVL tree

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CSE331 Lecture 20 Advanced Associative
Structures
Chapter 11 Binary Search tree, red-black tree,
and AVL tree Binary Search Trees 2-3-4
Tree Insertion of 2-3-4 tree Red-Black
Trees Converting 2-3-4 tree to Red-Black tree
Four Situations in the Splitting of a
4-Node left child of a BLACK parent P prior
to inserting node oriented left-left from G
using a single right rotation oriented
left-right from G after the color flip Building
a Red-Black Tree Red-Black Tree
Representation Summary Slide
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Binary Search Tree, Red-Black Tree and AVL Tree
Example
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Two Binary Search Tree Example

• Insertion sequence 5, 15, 20, 3, 9, 7, 12, 17,
  6, 75, 100, 18, 25, 35, 40

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2-3-4 Tree (node types)
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2-3-4 Tree Example
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2-3-4 Tree InsertionSplits 4-nodes top-down
After split, 7 is added to leaf following normal
search tree order
Inserting 7 into tree requires split of 4-node
before adding 7 as child (leaf or part of
existing leaf)
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3 6 9
9
3
2
10
4
8
2
10
7 8
4
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2-3-4 Tree Insertion Details

• 4-nodes must be split on the way down
• Split involves promotion of middle value into
  parent node
• New value is always added to existing 2-node or
  3-node
• 2-3-4 tree is always perfectly balanced
• Implementation is wasteful of space
• Each node must have space for 3 values, 1 parent
  pointer and 4 child pointers, even if not always
  used
• There is an equivalent red-black binary tree

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Building 2-3-4 Tree by Series of Insertions

• Insertion Sequence 2, 15, 12, 4, 8, 10, 25, 35,
  55, 11, 9, 5, 7

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Insertion Sequence 2, 15, 12, 4, 8, 10, 25, 35,
55, 11, 9, 5, 7
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Insertion Sequence 2, 15, 12, 4, 8, 10, 25, 35,
55, 11, 9, 5, 7
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Red Black Tree Properties

• The root is always BLACK
• A RED parent never has a RED child
• Every path from ROOT to an EMPTY subtree has the
  same black height ( of BLACK nodes)
• Black Height of tree is O(log2n)

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Red Black Tree Node Types

• BLACK parent with two RED children is equivalent
  of a 4-node
• A 4-node split is accomplished by flipping the
  colors to RED parent two BLACK children, with
  possible rotation if colors conflict
• BLACK parent with single RED child is equivalent
  of a 3-node
• New nodes are inserted a RED leaves

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Red-Black Trees binary equivalents of 2-3-4
trees
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Converting a 2-3-4 Tree to Red-Black Tree
(top-down)
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Building red black tree by repeated insertions

• Insert root
• Each successive insertion point is found in
  normal search tree order
• Any 4-nodes (black parent with two red children)
  is split going down into tree
• Requires color changes and possible rotations of
  subtrees
• New value is added as RED leaf
• If result is RED child of RED parent, then
  rotations are required to correct

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Situations in the Splitting of a 4-Node (A X B)
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Required actions to split 4-node (A X B) with
parent P grandparent G

• 4-node is either child of black parent
• Flip subtree colors ONLY (X red, A B black)
• 4-node is child of red parent (left-left)
• Flip subtree colors single right rotation about
  root
• 4-node is child of red parent (right-right)
• Flip subtree colors single left rotation about
  root
• 4-node is child of red parent (left-right)
• Double rotation (first left, then right) about X
  then color X black, G P red
• 4-node is child of red parent (right-left)
• Double rotation (first right, then left) about X
  then color X black, G P red

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Left child of a Black parent PSplit requires
only a Color Flip
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Example split prior to inserting 55 flip colors
(40,50,60) insert leaf
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RED parent, 4-node oriented left-left from G
(grandparent)
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Red Parent Oriented Left-Right From G
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Red Parent Oriented Left-Right From G
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Red Parent Oriented Left-Right From G
After second (right) rotation about X and
coloring X black, G P red
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Building A Red-Black Tree Insertions 2, 15,
12, 4, 8, 10, 25, 35
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Building A Red-Black Tree Insertions 2, 15, 12,
4, 8, 10, 25, 35
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rbnode Representation of Red-Black Tree
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Summary Slide 1
- 2-3-4 tree - a node has either 1 value
and 2 children, 2 values and 3 children, or 3
values and 4 children - construction of
2-3-4 trees is complex, so we build an
equivalent binary tree known as a red-black tree 
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Summary Slide 2
- red-black trees - Deleting a node from a
red-black tree is rather difficult.   -
the class rbtree, builds a red-black tree