## Left-leaning Red-Black Trees

I think this is the easiest algorithm to learn and code. Robert Sedgewick’s Java code is always succinct. I have learnt how to code Tries based on his code. Now his Red-black tree code is easy to remember.

http://www.cs.princeton.edu/~rs/talks/LLRB/LLRB.pdf

Though the properties of this type of tree are different from the original Red-black tree I think it is sufficient to remember this condensed code.

These are the three lines and some small methods that we should remember mainly.

```        if ( isRed(h.right) && !isRed(h.left)) h = rotateLeft(h);
if ( isRed(h.left) && isRed(h.left.left)) h = rotateRight(h);
if ( isRed(h.left) && isRed(h.right)) colorFlip(h);
```

### Final tree

I think the fundamental rules are these apart from the other Red-black tree rules.

1. Disallow right-leaning red nodes.
2. Disallow more than two consecutive red nodes.

## Red-Black Tree

I will refine this post as I code more methods. This is an attempt to learn this data structure and explain what I think I understand using pictures drawn using TEX. I use the tikz editor.

The algorithm code is based on pseudo-code from “Introduction to Algorithms” by Thomas H. Cormen et al. I use Java language features like enums etc. wherever possible.

### Sample TEX code

```[->,>=stealth',level/.style={sibling distance = 5cm/#1,
level distance = 1.5cm}]
\node [arn_n] {3}
child{ node [arn_r] {2}
child{ node [arn_nil] {NIL}}
}
child{ node [arn_r] {4}
child{ node [arn_nil] {NIL}}
}
;
```

### Insert

```public class RedBlack {

private final Node SENTINEL= new Node();
private Node root= new Node();

public RedBlack() {
root = SENTINEL;
}

enum Color {
BLACK,
RED};

class Node{

public void setColor(Color color) {
this.color = color;
}

private Color color = Color.BLACK;

Node left,right,p;
int key;
}

public void insert( RedBlack tree,
Node z){

Node y =  SENTINEL;
Node x =  tree.root;

while( x != SENTINEL){
y = x;
if ( z.key < x.key){
x = x.left;
}else
x = x.right;
}

z.p = y;

if( y == SENTINEL){
tree.root = z;
}else if ( z.key < y.key){
z = y.left;
}else{
z = y.right;
}
z.left = z.right = SENTINEL;
z.setColor(Color.RED);
}
}
```

I check the colors like this.

```    public void inOrder(Node x){

if( x != SENTINEL){
inOrder(x.left);
System.out.println(x.key + "[" + x.color + "]");
inOrder(x.right);
}
}
```

Now this tree has to be fixed so that it conforms to the rules that govern a Red-Black tree. So I code a method to fix it. This method is based on the pseudo-code from the same book. But I think there is a bug which I haven’t found. This tree after it is fixed looks like this.

The black height property is 2.

The black height property for the left branch is 2 but it is 3 for one of the paths on the right.

I may have to read some other book because finding this bug is going to be hard.

Actually the tree should be like this.