Analysis of Red-Black Tree

Because of the rules of the Red-Black tree, its height is atmost 2log(N + 1).

Meaning that it is a balanced tree

Time Analysis:

1. Finding any node will take O(log N) time.2. Inserting any node will take O(log N) time.3. Deleting any node will take O(log N) time.

What is a Red-Black Tree?

Other than being a Binary Search Tree it has these specialcoloring properties:

1. Every node is colored either red or black.2. The root is always black.3. If a node is red, its children must be black.

4. Every path from a node to a null reference must containthe same number of black nodes.

Yes rule #4 is somewhat confusing Don’t Panic we will go over it

Rule #4Every path from a node to a null reference must containthe same number of black nodes.

B

DA

C

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X Y1. Side X you count 1 blacknode to any null reference.

2. Side Y you count 1 blacknode to any null reference.

This Balance must bemaintained through out thetree.

More on Coloring1. On any path from the root to a leaf, red nodes must not be adjacent.

2. However, any number of black nodes may appear in a sequence.

3. Every time you add a node it begins as red.

4. All null references are defined to be black.

Now to Insert a node

Case 1a) x’s uncle is Red, and x is right child.

B

DA

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x

y

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New x

X represents the newly inserted node

Insertion

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x

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New x

Case 1b) x’s uncle is Red, and x is left child

Note: For all Cases when x is rootchange x to black.

InsertionCase 2) x’s uncle is Black, and x is a right child.

Note: Triangles representNull reference.

C

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Left rotationof A

InsertionCase 3) x’s uncle is Black, x is left child.

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Right rotation and recolor.

When Inserting a NodeRemember:1. Insert nodes one at a time, and after every Insertionbalance the tree.

2. Every node inserted starts as a Red node.

3. Consult the cases, Every time two Red nodes touchmust rebalance at that point.

4. The root will always be Black.

Deleting a NodeAnd you thought Inserting a Node was fun, lets try Deleting.

Rules:If we delete a Red node, tree is Still a Red-Black tree.

Assume we delete a Black node:Let x be the child of the deleted node and w be its sibling.

If (x is Red, change it to Black and stop)

Else If (x is Black, mark it double black and applythe following)

Deleting a NodeCase 1) x’s sibling is red.

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X stays at the same black height, Continue down the tree.

x

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Single rotation and Recolor.

Deleting a NodeCase 2a) x’s sibling is black, x’s parent is black

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Dx

Re-color w, Decrease x height by one and continue up tree.

wx

Deleting a NodeCase 2b) x’s sibling is black, x’s parent is red.

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Swap x’s parent and x’s, siblings color.Terminal case is Red-Black Tree.

xw

Deleting a NodeCase 3) x’s parent is either, x’s sibling is black, x’s siblings leftchild is red, x’s siblings right child is black.

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AEC

xw

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xw

X’s black height stays same, change to case 4.

Deleting a NodeCase 4) x’s parent is either, x’s sibling is black, x’s siblingsleft child is either, x’s siblings right child is red.

B

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B

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A C

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x w

Terminal case, tree is Red-Black tree.

Deleting a Node

To deleting a node that has two children, we replace thedeleted node with the smallest node on its right sub-tree.

You cannot use given cases unless the node to be deletedhas one or fewer children.

Java Demo

http://gauss.ececs.uc.edu/RedBlackMozilla/redblack.html

Great Demo that shows step by step Insertionand Deletion.

Also contains Java source code.

Review for QuizInsertion:1. When building a tree, insert One node at a time.2. After inserting a node, consult your cases and balancethe tree.

Deletion:3. After deleting a node, consult cases, and balance tree.

Remember:4. No two Red nodes can touch.5. All paths through tree must contain same number ofBlack nodes.6. Must always have Binary Search Tree balance.