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A red-black tree is a type of self-balancing binary search tree, a data structure used in computer science, typically used to implement associative arrays. The original structure was invented in 1972 by Rudolf Bayer who called them "symmetric binary B-trees", but acquired its modern name in a paper in 1978 by Leonidas J. Guibas and Robert Sedgewick. It is complex, but has good worst-case running time for its operations and is efficient in practice: it can search, insert, and delete in O(log n) time, where n is total number of elements in the tree.
## Terminology

## Properties

## Analogy to B-trees of order 4

## Applications and related data structures

## Operations

### Insertion

A red-black tree is a special type of binary tree, used in computer science to organize pieces of comparable data, such as numbers.

In red-black trees, the leaf nodes are not relevant and do not contain data. These leaves need not be explicit in computer memory — a null child pointer can encode the fact that this child is a leaf — but it simplifies some algorithms for operating on red-black trees if the leaves really are explicit nodes. To save memory, sometimes a single sentinel node performs the role of all leaf nodes; all references from internal nodes to leaf nodes then point to the sentinel node.

Red-black trees, like all binary search trees, allow efficient in-order traversal of elements provided that there is a way to locate the parent of any node. The search-time results from the traversal from root to leaf, and therefore a balanced tree, having the least possible tree height, results in O(log n) search time.

A red-black tree is a binary search tree where each node has a color attribute, the value of which is either red or black. In addition to the ordinary requirements imposed on binary search trees, the following additional requirements of any valid red-black tree apply:

- A node is either red or black.
- The root is black. (This rule is used in some definitions and not others. Since the root can always be changed from red to black but not necessarily vice-versa this rule has little effect on analysis.)
- All leaves are black, even when the parent is black (The leaves are the null children.)
- Both children of every red node are black.
- Every simple path from a node to a descendant leaf contains the same number of black nodes, either counting or not counting the null black nodes. (Counting or not counting the null black nodes does not affect the structure as long as the choice is used consistently.)

These constraints enforce a critical property of red-black trees: that the longest path from the root to a leaf is no more than twice as long as the shortest path from the root to a leaf in that tree. The result is that the tree is roughly balanced. Since operations such as inserting, deleting, and finding values requires worst-case time proportional to the height of the tree, this theoretical upper bound on the height allows red-black trees to be efficient in the worst-case, unlike ordinary binary search trees.

To see why these properties guarantee this, it suffices to note that no path can have two red nodes in a row, due to property 4. The shortest possible path has all black nodes, and the longest possible path alternates between red and black nodes. Since all maximal paths have the same number of black nodes, by property 5, this shows that no path is more than twice as long as any other path.

In many presentations of tree data structures, it is possible for a node to have only one child, and leaf nodes contain data. It is possible to present red-black trees in this paradigm, but it changes several of the properties and complicates the algorithms. For this reason, this article uses "null leaves", which contain no data and merely serve to indicate where the tree ends, as shown above. These nodes are often omitted in drawings, resulting in a tree which seems to contradict the above principles, but which in fact does not. A consequence of this is that all internal (non-leaf) nodes have two children, although one or both of those children may be null leaves.

Some explain a red-black tree as a binary search tree whose edges, instead of nodes, are colored in red or black, but this does not make any difference. The color of a node in this article's terminology corresponds to the color of the edge connecting the node to its parent, except that the root node is always black (property 2) whereas the corresponding edge does not exist.

A red-black tree is similar in structure to a B-tree of order 4, where each node can contain between 1 to 3 values and (accordingly) between 2 to 4 child pointers. In such B-tree, each node will contain only one value matching the value in a black node of the red-black tree, with an optional value before and/or after it in the same node, both matching an equivalent red node of the red-black tree.

One way to see this equivalence is to "move up" the red nodes in a graphical representation of the red-black tree, so that they align horizontally with their parent black node, by creating together a horizontal cluster. In the B-tree, or in the modified graphical representation of the red-black tree, all leaf nodes are at the same depth.

The red-black tree is then structurally equivalent to a B-tree of order 4, with a minimum fill factor of 33% of values per cluster with a maximum capacity of 3 values.

However, the conversion of the structurally equivalent B-tree back to a red-black tree can produce multiple results. Effectively, if a B-tree cluster contains only 1 value, the minimum (and has two child pointers), then this value will be black in the red-black tree. If the cluster contains 3 values, then the central value will be black and each value stored on its sides will be red in the red-black tree. However if the cluster contains two values, each one can become the black node in the red-black tree (and the other one will be red).

So the order-4 B-tree does not maintain which of the values contained in each cluster is the root black tree for the whole cluster and the parent of the other values in the same cluster. Despite this, the operations on red-black trees are more economical because you don't have to maintain the vector of values (it may be costly if values are stored directly in each node rather than being stored by simple reference. But B-tree nodes are more economical in space, because you don't need to store the colors attribute for each node (instead you have to know which slot in the vector of 3 values of the cluster is used; if values are stored by references, for example they are objects, you just need to be able to make a distinction with null references, and so the cluster is just represented by a vector containing 3 slots for value pointers plus 4 slots for child references in the tree). In that case, the B-tree can be more compact in memory, it will increase the data locality.

The same analogy can be made with B-trees with larger orders that can be structurally equivalent to a colored binary tree: you just need more colors. Suppose that you add blue, then the blue-red-black tree defined like red-black trees but with the additional constraint that no two successive nodes in the hierarchy will be blue and all blue nodes will be children of a red node, then it becomes equivalent to a B-tree whose clusters will have at most 7 values in the following colors: blue, red, blue, black, blue, red, blue (For each cluster, there will be at most 1 black node, 2 red nodes, and 4 blue nodes).

For moderate volumes of values, insertions and deletions in a colored binary tree are faster compared to B-trees because colored trees don't attempt to maximize the fill factor of each horizontal cluster of nodes (only the minimum fill factor is guaranteed in colored binary trees, limiting the number of splits or junctions of clusters). B-trees will be faster for performing rotations (because rotations will frequently occur within the same cluster rather than with multiple separate nodes in a colored binary tree). However for storing large volumes, B-trees will be much faster as they will be more compact by grouping several children in the same cluster where they can be accessed locally.

All optimizations possible in B-trees to increase the average fill factors of clusters are possible in the equivalent multicolored binary tree. Notably, maximizing the average fill factor in a structurally equivalent B-tree is the same as reducing the total height of the multicolored tree, by increasing the number of non-black nodes. The worst case occurs when all nodes in a colored binary tree are black, the best case occurs when only a third of them are black (and the other two thirds are red nodes).

Red-black trees offer worst-case guarantees for insertion time, deletion time, and search time. Not only does this make them valuable in time-sensitive applications such as real-time applications, but it makes them valuable building blocks in other data structures which provide worst-case guarantees; for example, many data structures used in computational geometry can be based on red-black trees.

The AVL tree is another structure supporting O(log n) search, insertion, and removal. It is more rigidly balanced than Red-Black trees, leading to slower insertion and removal but faster retrieval.

Red-black trees are also particularly valuable in functional programming, where they are one of the most common persistent data structures, used to construct associative arrays and sets which can retain previous versions after mutations. The persistent version of red-black trees requires O(log n) space for each insertion or deletion, in addition to time.

Red-black trees are an isometry of 2-3-4 trees. In other words, for every 2-3-4 tree, there is a corresponding red-black tree with data elements in the same order. The insertion and deletion operations on 2-3-4 trees are also equivalent to color-flipping and rotations in red-black trees. This makes 2-3-4 trees an important tool for understanding the logic behind red-black trees, and this is why many introductory algorithm texts introduce 2-3-4 trees just before red-black trees, even though 2-3-4 trees are not often used in practice.

In 2008, Sedgewick introduced a simpler version of red-black trees called Left-Leaning Red-Black Trees by eliminating a previously unspecified degree of freedom in the implementation. This page, however, describes the 1978 version.

Read-only operations on a red-black tree require no modification from those used for binary search trees, because every red-black tree is a special case of a simple binary search tree. However, the immediate result of an insertion or removal may violate the properties of a red-black tree. Restoring the red-black properties requires a small number (O(log n) or amortized O(1)) of color changes (which are very quick in practice) and no more than three tree rotations (two for insertion). Although insert and delete operations are complicated, their times remain O(log n).

Insertion begins by adding the node as we would in a simple binary search tree and coloring it red. What happens next depends on the color of other nearby nodes. The term uncle node will be used to refer to the sibling of a node's parent, as in human family trees. Note that:

- Property 3 (All leaves, including nulls, are black) always holds.
- Property 4 (Both children of every red node are black) is threatened only by adding a red node, repainting a black node red, or a rotation.
- Property 5 (All paths from any given node to its leaf nodes contain the same number of black nodes) is threatened only by adding a black node, repainting a red node black, or a rotation.

- Note: The label N will be used to denote the node being inserted; P will denote N's parent node, G will denote N's grandparent, and U will denote N's uncle. Note that in between some cases, the roles and labels of the nodes are exchanged, but in each case, every label continues to represent the same node it represented at the beginning of the case. Any color shown in the diagram is either assumed in its case or implied by these assumptions.

struct node * uncle(struct node *n) { struct node *g = grandparent(n);

if (g == NULL)

return NULL; // No grandparent means no uncleif (n->parent == g->left) return g->right; else return g->left; }

Case 1: The new node N is at the root of the tree. In this case, it is repainted black to satisfy Property 2 (The root is black). Since this adds one black node to every path at once, Property 5 (All paths from any given node to its leaf nodes contain the same number of black nodes) is not violated.

Case 2: The new node's parent P is black, so Property 4 (Both children of every red node are black) is not invalidated. In this case, the tree is still valid. Property 5 (All paths from any given node to its leaf nodes contain the same number of black nodes) is not threatened, because the new node N has two black leaf children, but because N is red, the paths through each of its children have the same number of black nodes as the path through the leaf it replaced, which was black, and so this property remains satisfied.

- Note: In the following cases it can be assumed that N has a grandparent node G, because its parent P is red, and if it were the root, it would be black. Thus, N also has an uncle node U, although it may be a leaf in cases 4 and 5.

Case 3: If both the parent P and the uncle U are red, then both nodes can be repainted black and the grandparent G becomes red (to maintain Property 5 (All paths from any given node to its leaf nodes contain the same number of black nodes)). Now, the new red node N has a black parent. Since any path through the parent or uncle must pass through the grandparent, the number of black nodes on these paths has not changed. However, the grandparent G may now violate properties 2 (The root is black) or 4 (Both children of every red node are black) (property 4 possibly being violated since G may have a red parent). To fix this, this entire procedure is recursively performed on G from case 1. Note that this is a tail-recursive call, so it could be rewritten as a loop; since this is the only loop, and any rotations occur after this loop, this proves that a constant number of rotations occur. |

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Last updated on Saturday September 27, 2008 at 11:44:03 PDT (GMT -0700)

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Last updated on Saturday September 27, 2008 at 11:44:03 PDT (GMT -0700)

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