Red Black Tree Essentials
Notes from “Introduction to
Algorithms”, Cormen et al.
Definition
Red Black Trees
A red-black tree is a binary search tree with an extra bit of
storage per node.
The extra bit represents the color of the node. It's either red
or black.
Each node contains the fields: color, key, left, right, and p. Any
nil pointers are regarded as pointers to external nodes (leaves)
and key bearing
nodes are considered as internal nodes of the tree.
See the video at:
http://www.youtube.com/watch?v=vDHFF4wjWYU
Properties
Red-black tree properties:
1. Every node is either red or black.
2. The root is black.
3. Every leaf (nil) is black.
4. If a node is red then both of its children are
black.
5. For each node, all paths from the node to
descendant leaves contain the
same number of black nodes.
Implications
• From these properties, it can be shown
(with a proof by induction) that the tree has a
height no more than 2 * Lg(n + 1).
• Thus, worst case lookUp, insert, delete are all
Θ(Log n).
Three essential methods
• Rotation
• Insertion
• Insert-fixUp
LeftRotate(T,x)
pre: right[x] != nil[T]
pre: root's parent is nill[T]
Left-Rotate(T,x)
y = right[x]
right[x] = left[y]
p[left[y]] = x
p[y] = p[x]
if p[x] == nil[T] then root[T] = y
else
if x == left[p[x]] then left[p[x]] = y
else
right[p[x]] = y
left[y] = x
p[x] = y
Red Black Insert
RB-Insert(T,z)
y = nil[T]
x = root[T]
while x != nil[T]
y=x
if key[z] < key[x] then
x = left[x]
else
x = right[x]
p[z] = y
if y = nil[T]
root[T] = z
else
if key[z] < key[y] then
left[y] = z
else
right[y] = z
left[z] = nil[T]
right[z] = nil[T]
color[z] = RED
RB-Insert-fixup(T,z)
RB-Insert-fixup(T,z)
RB-Insert-fixup(T,z) {
while(z's parent is Red) {
set y to be z's uncle
if uncle y is Red {
color parent and uncle black
color grandparent red
set z to grandparent
}
else { // the uncle is black
if (zig zag) { // make it a zig zig
set z to parent
rotate to zig zig
}
// rotate the zig zig and finish
color parent of z black
color grandparent of z red
rotate grand parent of z
}
} // end while
color root black
}