Red–black tree

In computer science, a red–black tree is a specialised binary search tree data structure noted for fast storage and retrieval of ordered information, and a guarantee that operations will complete within a known time. Compared to other self-balancing binary search trees, the nodes in a red-black tree hold an extra bit called "color" representing "red" and "black" which is used when re-organising the tree to ensure that it is always approximately balanced.

Red–black tree
Type	Tree
Invented	1978
Invented by	Leonidas J. Guibas and Robert Sedgewick
Complexities in big O notation
Space complexity Space ${\displaystyle O(n)}$ Time complexity Function Amortized Worst Case Search ${\displaystyle O(\log n)}$ ${\displaystyle O(\log n)}$ Insert ${\displaystyle O(1)}$ ${\displaystyle O(\log n)}$ Delete ${\displaystyle O(1)}$ ${\displaystyle O(\log n)}$

When the tree is modified, the new tree is rearranged and "repainted" to restore the coloring properties that constrain how unbalanced the tree can become in the worst case. The properties are designed such that this rearranging and recoloring can be performed efficiently.

The (re-)balancing is not perfect, but guarantees searching in Big O time of ${\displaystyle O(\log n)}$, where ${\displaystyle n}$ is the number of entries (or keys) in the tree. The insert and delete operations, along with the tree rearrangement and recoloring, are also performed in ${\displaystyle O(\log n)}$ time.

Tracking the color of each node requires only one bit of information per node because there are only two colors. The tree does not contain any other data specific to it being a red–black tree, so its memory footprint is almost identical to that of a classic (uncolored) binary search tree. In some cases, the added bit of information can be stored at no added memory cost.