The 2–4 Trees

General red-black tree

Here, we present red-black trees, a version of binary search trees with logarithmic height. Red-black trees are one of the most widely used data structures. They appear as the primary search structure in many library implementations, including the Java Collections Framework and several implementations of the C++ Standard Template Library. They are also used within the Linux operating system kernel. There are several reasons for the popularity of red-black trees:

1. A red-black tree storing n values has height at most $2 \log n.$
2. The add(x) and remove(x) operations on a red-black tree run in $O(\log n)$ worst-case time.
3. The amortized number of rotations performed during an add(x) or remove(x) operation is constant.

The first two of these properties already put red-black trees ahead of skiplists, treaps, and scapegoat trees. Skiplists and treaps rely on randomization and their $O(\log n)$ running times are only expected. Scapegoat trees have a guaranteed bound on their height, but add(x) and remove(x) only run in $O(\log n)$ amortized time. The third property is just icing on the cake. It tells us that that the time needed to add or remove an element $x$ is dwarfed by the time it takes to find $x$.

Note: The skiplists and treaps also have the third property in the expected sense.

However, the nice properties of red-black trees come with a price: implementation complexity. Maintaining a bound of $2 \log n$ on the height is not easy. It requires a careful analysis of a number of cases. We must ensure that the implementation does exactly the right thing in each case. One misplaced rotation or change of colour produces a bug that can be very difficult to understand and track down.

Rather than jumping directly into the implementation of red-black trees, we will first provide some background on a related data structure: 2-4 trees. This will give some insight into how red-black trees were discovered and why efficiently maintaining them is even possible.