# Red Black Trees (under develpment)ΒΆ

The algorithms shown in Implementing a 2 3 Tree in C++14 and Implementing a 2 3 4 Tree in C++14 ensure a tree structure that is always balanced, but both a 2-3 tree and 2-3-4 tree waste storage because their 3-node and 4-nodes are not always full.

A red black tree is a way of representing a 2 3 4 tree as a nearly‘balanced binary search tree. Each edge of a red black tree is colored red or black, where red edges represent the values within a 3- or 4-node, while black edges represent the children. Using these coloring conventions, the nodes of a 2 3 4 tree map map to the nodes of a red black tree as shown in the image below (taken from these 2 3 4 Tree Lecture Notes):

If both edges (children) of a node are black, the node represents a 2-node. If one child is red and the other is black, the node represents a 3-node. If both children are red, then that node represents a 4-node. There is only one representation of a 2 3 4 tree’s 2-node, and there is only one possible representation of 2 3 4 tree’s 4-node, but there are two possible representations of a 3-node, but both are symmetrical, and both represent a unique 4-node.

A red black tree also conforms to these invariants:

- The root is always black, i.e., it has two black edges. It thus represents a 2-node.
- black condition - every path from the root to a leaf node has the same number of black links/nodes
- red condition - no path from the root a leaf node has two or more consecutive red links/nodes (ie. every red node will have a black parent)

Todo

How do these invariants map a 2 3 4 tree? What is there purpose?

Other sources:

- Trees.
- Mapping 2 3 4 Trees to Red Black Trees.
- USC lecture slides on B-Trees (2-3, 2 - 3 - 4) and Red/Black Trees.
- Lecture on Red-Black Trees. Discussion of red black trees starts on p. 44. Also see this University of Michigan