1

More Trees II

2-3-4 Trees, Red-Black Trees, B Trees

2

Objectives

You will be able to

Describe Red-Black trees. Describe B-Trees.

3

Red-Black Trees

A way of constructing 2-3-4 trees from 2-nodes.

Defined as a BST which: Has two kinds of links (red and black). Every path from root to a leaf node has same

number of black links. No path from root to leaf node has more than

two consecutive red links.

Data member added to store color of link from parent.

4

Red-Black Trees

enum ColorType {RED, BLACK};

class RedBlackTreeNode{public: DataType data; ColorType parentColor; // RED or BLACK RedBlackTreeNode * parent; RedBlackTreeNode * left; RedBlackTreeNode * right;}

Now possible to construct a red-black tree to represent a 2-3-4 tree

5

Red-Black Trees

We will convert each 3-Node in the 2-3-4 tree into two 2-Nodes.

Each 4-Node into three 2-Nodes.

6

Red-Black Trees

2-3-4 tree represented by red-black trees as follows:

Make a link black if it is a link in the 2-3-4 tree. Make a link red if it connects nodes containing values in

same node of the 2-3-4 tree.

Some authors use h and v instead of red and black.

h (horizontal) links connect nodes from the same node of the 2-3-4 tree.

v (vertical) links are links from the 2-3-4 tree.

7

Example

2-3-4 tree

Corresponding red-black tree

55

59

8

Adding to a Red-Black Tree

Do top-down insertion as with 2-3-4 tree

1. Search for place to insert new node – Keep track of parent, grandparent, great grandparent.

2. When 4-node q encountered, split as follows:a. Change both links of q to blackb. Change link from parent to red:

9

Adding to a Red-Black Tree

3. If now two consecutive red links, (from grandparent gp to parent p to q)Perform appropriate AVL type rotation determined by direction (left, right, left-right, right-left) from gp -> p-> q

10

Example

Let's convert the quiz solution into a Red-Black tree.

After Adding WY

GA

DE IL IN RI

PA TX

MA

VT WY MI NY OH

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Corresponding Red-Black Tree

GA

DE RI

PA

MA

WYOH

TX

IL IN VT NYMI

12

Add NM

GA

DE RI

PA

MA

WYOH

TX

IL IN VT NYMI

Will be right child of MI

We have to split the "4-node" of MI-NY-OH

13

After Adding NM

GA

DE RI

PA

MA

WYOH

TX

IL IN VT

NY

MI NM

End of Section14

B-Trees

Drozdek Chapter 7

15

16

B-Trees

Etymology unknown

Rudolf Bayer and Ed McCreight invented the B-tree while working at Boeing Research Labs in 1971, but they did not explain what, if anything, the B stands for.

Balanced Trees ? Bayer Trees ? Boeing Trees ?

See http://en.wikipedia.org/wiki/B-tree

Many variations: B+ Trees, B* Trees See Drozdek Chapter 7

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B-Trees Previous trees used in internal searching schemes

Tree sufficiently small to be all in memory

B-trees are intended for external searching Data stored in secondary memory Each node is a block on disk. Typically the "data" in a node is really a pointer.

B-tree of order m has properties: The root has at least two subtrees unless it is a leaf. Each node stores at most m – 1 data values

and has at most m links to subtrees. Each internal node stores at least ceil(m/2) data values. All leaves on same level

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B-Trees

A 2-3-4 tree is a B-tree of order 4 Note example below of order 5 B-tree

Best performance for disk storage found to be with values for 50 ≤ m ≤ 400

End of Presentation