A red-black tree is a binary tree representation of a 2-3-4 tree. The child pointer of a node is either red or black. If the child pointer was present in the original 2-3-4 tree, it is a black pointer; otherwise, it is a red pointer.

A 2-, 3- and 4-nodes is transformed into its red-black representation as follows:

# Red Black Trees (under develpment)¶

**Contents Under Development...**

The algorithms shown in Implementing a 2 3 4 Tree in C++17 ensure a tree structure that is always balanced, but a 2-3-4 tree wastes storage because its 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.

“Every 2-, 3-, and 4-node in a 2 3 4 tree can be converted to at least 1 black node and 1 or 2 red children of the black node. Red nodes are always ones that would join with their parent to become a 3- or 4-node in a 2-3-4 tree” from http://ee.usc.edu/~redekopp/cs104/slides/L19b_BalancedBST_BTreeRB.pdf slide 45.

The crucial slides, notes, and explanations of red black trees and how to transform the nodes of a 2-3-4 tree into red black tree:

- This Open Data structures article explains how the 2 3 4 algorithms map to the red black tree algorithms.
- These Red black lecture notes are the basis for a solid introduction in red black trees. They use use 2 3 4 trees as the basis for understanding red black trees. They seem conceptually thorough, in depth and tutoriall-oriented. It has succinct proofs about 2 3 4 tree and red black tree equivalence–I believe.
- The Standford CS166 page has excellent slides on
**Balanced Trees, Part I**and**Balaaced Trees, Part 2**that are very good. They explain how a more memory efficient BST tree than multiway trees like 2 3 4 trees motivated the inventation of red black trees. It shows the isometry between 2 3 4 tree4 and red black trees and how the insertion and deletion algorithm that maintain a balanced 2 3 4 tree are implemented in a red black tree, something that involves many special cases. - Mapping 2-3-4 Trees into Red-Black Trees shows both the mapping from 2 3 4 trees to red black trees, and how splitting 4-nodes works in a red black tree.
- National Chung University pdf shows how to transform a 2 3 4 tree into a red black tree starting at slide 67 and following. It makes more sense than the USC slides.

Other links of particular value are:

- This stackoverflow explanation has an excellent illustration of how 2 3 4 trees map to red black trees. It is very good.
- These Scribd slides, especially the last half, really explain red black trees as a type of 2 3 4 trees, and how how the operations of 2 3 4 tree map to red black trees.
- These Digipen.edu slides give an overview of all types of trees–BST, 2-3 tree, and red black trees–and the general concept of rotations

These linkd discuss insertion cases:

- http://www.cs.tulane.edu/~carola/teaching/cmps2200/fall17/slides/RB-trees.pdf
- https://www.cs.utexas.edu/~scottm/cs314/handouts/slides/Topic23RedBlackTrees.pdf
- https://www.cs.utexas.edu/~scottm/cs314/handouts/slides/Topic23RedBlackTrees.pdf

NIU also has a very succinct illustration and explanation of how 2-, 3- and 4-nodes correspond to red and black nodes and how describes the varies insertion senarios.

## Thoughts so far¶

A red-black tree is a binary tree representation of a 2-3-4 tree. The 2- and 4-nodes have only one equiavlent representation in a red black tree, but a 3-node can be represented two possilbe ways. <Describe node coloring in reb black trees> The nodes of a 2-, 3- and 4-nodes ares transformed into red-black tree nodes as follows:

A 2-node

a 4-node

a 3-node

Ulitmately explain why the invarient of the red black tree always holds true under the mappings described above.

A red black tree corresponds to a unique 2-3-4 tree; however, that 2-3-4 tree can be represented by different red black trees, but for all “tallest leaf” - “shortest” leaf <= 2 (I need the correct term for tallest and shortest leaf).

Todo

Take one of the trees outputted by my 2 3 4 tree test code and convert it into a red black tree.

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