Lab tomorrow

Drawing red-black trees

Bring two colors of pens

Red-Black Trees

Another way to balance binary search trees

Chapter 28.30 in Carrano/Savitch

What is a red-black tree

• It is a binary search tree in which each node is either red or black

• The root is always black

• If a node is red, its children must be black

• Every path from a node to NULL pointer must contain the same number of black nodes

– The paths we are interested in are to a NULL pointer

– This means the path from the root to items with no children (a leaf), or to items with one child

• Note that the paths we are interested in are not necessarily to a leaf!

Red-black tree definition

• A red-black tree is a BST that is empty, or in which the root node is colored black, and every other node is colored red or black, and the following properties are satisfied:

• Red rule: If an node is red, it parents must be black

– i.e. no two red nodes together

• Path rule: The number of black nodes must be the same in all paths from the root to a node with no children or with one child.

– Another way to say this is from the root node to a node with a NULL pointer

• Look at some BST on the overhead and decide which are red-black trees

Inserting into a red-black tree

• The new item is placed as a leaf in the tree

• If the new node is colored black, it will violate the path rule

– So all new insertions are red

• If the parent is black we are done.

• If the parent is red, we would violate the red rule

– So the parent being red indicates the need to adjust the tree

– This adjustment is done by color changes and tree rotations

Insertion considerations

• To decide what needs to be done after insertions, we look at the local neighborhood. This consists of the

• curr_node the node being inserted

• parent of node being inserted

• grandparent of the node being inserted

• uncle the sibling of parent

– Since an adjustment may involve recoloring, it may just move the problem two steps up the tree

– In this case it is dealt with recursively

Inserting nodes into a red-black tree

• Find the place to insert into the tree, and color the new node red

– If the parent is black, we are done

– If it is red, an adjustment must be performed

• Six cases dealing with parent and uncle

New node Uncle Parent

• Either red red left child

• Right black red left child

• Left black red left child

• The other three cases are mirror images with the parent being a red right child

Case 1

CurrNode— either left or right child Parent – red left child Uncle– red

What to do:

Recolor parent, uncle, grandparent

may have to readjust if result makes two reds together higher in the tree

Case 2– ziz-zag case

CurrNode—right child Parent – red left child Uncle– black or none

There is a ziz-zag between the current node, the parent

What to do:

Rotate parent and CurrNode and reset CurrNode

It then becomes case 3

Case 3

CurrNode—left child Parent – red left child Uncle– black or none

What to do:

Change color of parent and grandparent

Rotate(parent, grandparent)

Insert into a red-black tree

53, 27, 75, 25, 70, 41, 38, 16, 59, 36

23, 15, 40, 8, 17, 21, 18, 16, 55

Efficiency of Red-Black trees

• A red-black tree has just one extra bit of storage in each node, its color

• Inserting a node runs in O(log n) time since the rotations take constant time

• Deleting a node is only slightly more complex than inserting

– It also runs in O(log n) time

Other trees to practice

• 25, 27, 16, 36, 38, 53, 41, 59, 75, 70

Summary

Case 1 uncle—red; parent -- red left child; CurNode -- either L or R

Recolor parent, uncle, grandparent

Case 2 CurNode--right child; parent--a red left child, uncle – black or none

– Rotate (parent, CurNode) and reset CurNode

– It then becomes case 3

Case 3 CurNode -- left child; parent--red left child; uncle–black or none

– Change color of parent and grandparent

– Rotate(parent, grandparent)