Fundamentals of Red-Black Trees

Red-black trees and 2-4 trees

At first it might seem surprising that a red-black tree can be efficiently updated to maintain the black-height and no-red-edge properties, and it seems unusual to even consider these as useful properties. However, red-black trees were designed to be an efficient simulation of 2-4 trees as binary trees.

Refer to the below figure:

Consider any red-black tree, $T$, having n nodes and perform the following transformation: Remove each red node u and connect u's two children directly to the (black) parent of u. After this transformation, we are left with a tree $T’$ having only black nodes.

Every internal node in $T’$ has two, three, or four children: A black node that started out with two black children will still have two black children after this transformation. A black node that started out with one red and one black child will have three children after this transformation. A black node that started out with two red children will have four children after this transformation. Furthermore, the black-height property now guarantees that every root-to-leaf path in $T’$ has the same length. In other words, $T’$ is a 2-4 tree!

The 2-4 tree ...