# Red-Black Trees

Binary search trees work best when they are balanced or the path length from root to any leaf is within some bounds. The red-black tree algorithm is a method for balancing trees. The name derives from the fact that each node is colored red or black, and the color of the node is instrumental in determining the balance of the tree. During insert and delete operations, nodes may be rotated to maintain tree balance. Both average and worst-case search time is O(lg n). For details, consult Cormen [1990].

## Theory

A red-black tree is a balanced binary search tree with the following properties:
1. Every node is colored red or black.
2. Every leaf is a NIL node, and is colored black.
3. If a node is red, then both its children are black.
4. Every simple path from a node to a descendant leaf contains the same number of black nodes.
The number of black nodes on a path from root to leaf is known as the black-height of a tree. These properties guarantee that any path from the root to a leaf is no more than twice as long as any other. To see why this is true, consider a tree with a black height of two. The shortest distance from root to leaf is two, where both nodes are black. The longest distance from root to leaf is four, where the nodes are colored (root to leaf): red, black, red, black. It is not possible to insert more black nodes as this would violate property 4, the black-height requirement. Since red nodes must have black children (property 3), having two red nodes in a row is not allowed. The largest path we can construct consists of an alternation of red-black nodes, or twice the length of a path containing only black nodes. All operations on the tree must maintain the properties listed above. In particular, operations which insert or delete items from the tree must abide by these rules.

## Insertion

To insert a node, we search the tree for an insertion point, and add the node to the tree. A new node replaces an existing NIL node at the bottom of the tree, and has two NIL nodes as children. In the implementation, a NIL node is simply a pointer to a common sentinel node that is colored black. After insertion, the new node is colored red. Then the parent of the node is examined to determine if the red-black tree properties have been violated. If necessary, we recolor the node and do rotations to balance the tree.

By inserting a red node with two NIL children, we have preserved black-height property (property 4). However, property 3 may be violated. This property states that both children of a red node must be black. Although both children of the new node are black (they're NIL), consider the case where the parent of the new node is red. Inserting a red node under a red parent would violate this property. There are two cases to consider:

• Red parent, red uncle: Figure 3-6 illustrates a red-red violation. Node X is the newly inserted node, with both parent and uncle colored red. A simple recoloring removes the red-red violation. After recoloring, the grandparent (node B) must be checked for validity, as its parent may be red. Note that this has the effect of propagating a red node up the tree. On completion, the root of the tree is marked black. If it was originally red, then this has the effect of increasing the black-height of the tree.

• Red parent, black uncle: Figure 3-7 illustrates a red-red violation, where the uncle is colored black. Here the nodes may be rotated, with the subtrees adjusted as shown. At this point the algorithm may terminate as there are no red-red conflicts and the top of the subtree (node A) is colored black. Note that if node X was originally a right child, a left rotation would be done first, making the node a left child.
Each adjustment made while inserting a node causes us to travel up the tree one step. At most 1 rotation (2 if the node is a right child) will be done, as the algorithm terminates in this case. The technique for deletion is similar.

## Implementation

An ANSI-C implementation for red-black trees is included. Typedef T and comparison operators compLT and compEQ should be altered to reflect the data stored in the tree. Each Node consists of left, right, and parent pointers designating each child and the parent. The node color is stored in color, and is either RED or BLACK. The data is stored in the data field. All leaf nodes of the tree are sentinel nodes, to simplify coding. The tree is based at root, and initially is a sentinel node.

Function insertNode allocates a new node and inserts it in the tree. Subsequently, it calls insertFixup to ensure that the red-black tree properties are maintained. Function deleteNode deletes a node from the tree. To maintain red-black tree properties, deleteFixup is called. Function findNode searches the tree for a particular value.