More Trees II
2-3-4 Trees, Red-Black Trees, B Trees
1
Objectives
You will be able to


Describe Red-Black trees.
Describe B-Trees.
2
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.
3
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
4
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.
5
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.
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Example

2-3-4 tree

Corresponding red-black tree
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Adding to a Red-Black Tree

1.
2.
Do top-down insertion as with 2-3-4 tree
Search for place to insert new node – Keep
track of parent, grandparent, great
grandparent.
When 4-node q encountered, split as follows:
a. Change both links of q to black
b. Change link from parent to red:
8
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
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Example

Let's convert the quiz solution into a
Red-Black tree.
10
After Adding WY
MA
GA
DE
IL IN
PA TX
MI NY OH
RI
VT WY
11
Corresponding Red-Black Tree
MA
GA
DE
IL
IN
TX
PA
MI
NY
OH
RI
VT
WY
12
Add NM
MA
GA
DE
IL
IN
TX
PA
MI
NY
OH
RI
VT
WY
Will be right child of MI
We have to split the "4-node" of MI-NY-OH
13
After Adding NM
MA
GA
DE
IL
IN
NY
MI
NM
TX
PA
OH
RI
VT
WY
End of Section
14
B-Trees

Drozdek Chapter 7
15
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


B-trees are intended for external searching




Tree sufficiently small to be all in memory
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
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