Red-Black Trees

Niño GuerreroDexter Paul Gumahad

Rene MaderaReymart Pagente

Data Structure || JRMSU Computer Science

Red-Black Trees

• A Red-Black tree is a binary search tree with these traits:– Every node is either red or black.– The root is always black.– If a node is red, its children must be black.– Every path from any node to a null must have the

same number of black nodes.

• Operations on Red-Black trees are O(log N).

Data Structure || JRMSU Computer Science

c

a f

b gd

e

Red-Black tree

Data Structure || JRMSU Computer Science

Red-Black Trees

• Red-black trees can be thought of as binary tree representations of a 2-3-4 tree.

• A 2-3-4 tree is a B-tree of degree 4.• This means each internal node has either 2, 3,

or 4 children.• The next slide illustrates this.

Data Structure || JRMSU Computer Science

c f

a b d e g

2-3-4 tree

Data Structure || JRMSU Computer Science

c

a f

b gd

e

c f

a b d e g

How the Trees Relate

Each black node represents a node in the 2-3-4 tree.Each red node is in the same node with its black parent.This explains why every path has the same number of black nodes.

Data Structure || JRMSU Computer Science

Mappings

a b

a

b a

b

or

a b c

b

ca

aa

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Inserting into a Red-Black Tree

• An inserted node is placed as in a binary search tree as a red node, unless it is the root.

• If the parent of the new node is also red, called a “double red”, the tree will have to be adjusted, since a red child must have a black parent.

Data Structure || JRMSU Computer Science

Two Casesa

c

bz

w

a

c

bz

wv

u

1.

2.

Inserting node z causes double red, and z’s parent has black sibling w. This occurs when a 4-node is malformed: must restructure.

Inserting node z causes double red, and z’s parent has red sibling w. This occurs when a 4-node overflows: must recolor.

Note: these are partial trees so not allof their subtrees are shown.

v

u

Data Structure || JRMSU Computer Science

a

c

b

Restructuring

a c

b

w

c

a

b

w

a

b

c

w

c

b

a

w

Double rotation Single rotation

Parent is black,children are red.

The malformed 4-node isnow a well-formed 4-node.

Data Structure || JRMSU Computer Science

Recoloring

b

d

c

w a

b

c

d

w a

b

d

c

w a

b

c

d

w a

Note: since b is now red, this can cause a double red with b’s parent.If so, the two cases are considered again for b and its parent.Data Structure || JRMSU Computer Science

Example

e e

h

e

h

k

1. Insert e 2. Insert h 3. Insert k, h hasnull (black) sibling

e

h

k

4. Restructure

e

h

k

5. Insert n, k hasred sibling

n

e

h

k

n

6. Recolor (root left black)Data Structure || JRMSU Computer Science

End

Data Structure || JRMSU Computer Science