nodes can have multiple items in them and can have anywhere from 2 to not? we can keep them complete. - (2,4) trees are fun! could find a way to keep a tree as complete (or close to complete) by node? A red–black tree is similar in structure to a B-tree of order 4, where each node can contain between 1 and 3 values and (accordingly) between 2 and 4 child pointers. The delete method for a red-black tree is more complicated than the insert method and is not included in these notes. of a binary search tree. A Red-Black tree is a binary search tree in which each node is colored red or black! always be balanced! You will not be responsible for knowing the details of how to delete a key from a red-black tree. A 2-4 tree is a search tree with special kinds of nodes. The delete operation is similar in feel to the insert operation, but more complicated. leaves to the root, or O(log(n)). But what Interesting excercise: what is the BEST height of the tree? Restructuring is O(1) since it involves changing at most five pointers to tree nodes. keys(m) → (k > leftKey(n)) ∧ (k < rightKey(n))), ∀m,n RightDescendant(m,n) → (∀ k ∈ keys(m) → k > rightKey(n)), ∀m,n MiddleLeftDescendant(m,n) → (∀ k ∈ But what if we go to add and item, and the node is full, then When we promote, we can perform a transfer with? Red nodes represent the extra keys in 3-nodes and 4-nodes! The child node has two children of its own: either W and X or X and Y. If that is the case, then we recursively handle the double-red situation starting at G and G's parent (instead of K and K's parent). each node has a color (red or black) associated with it (in addition to its key and left and right children), determine whether a key value is in the tree (, print all of the key values in sorted order (, restore red-black tree properties (if necessary). The BST insert operation is O(height of tree) which is O(log N) because a red-black tree is balanced. These nodes can have multiple items in them and can have anywhere from 2 to 4 children. Although it is not intuitively obvious, the algorithms for restoring these properties after a node has been added or removed result in a tree that stays balanced. keys(m) → (k > leftKey(n)) ∧ (k < middleKey(n))), ∀m,n MiddleRightDescendant(m,n) → (∀ k ∈ keys(m) → k > rightKey(n)). In the next section, we discuss the mapping of a 2-3-4 tree to a red-black tree. Here we will look at yet another kind of balanced tree called
The second two possibilites are mirror-images of the first two possibilites: Once a restructuring is done, the double-red situation has been handled and there's nothing more to do (you should convince yourself, by looking at the diagrams above, that restructuring will not result in a violation of the black property). Let P be K's parent. Red-black trees are binary search trees that store one additional piece of information in each node (the node's color) and satisfy three properties. However, adding a red leaf node may affect T's satisfaction of the red property, so we will need to check if that is the case and, if so, fix it (step 3). You can interactively add and remove items from the red-black tree and see the corresponding changes in the 2-4 tree: a balanced tree -- a tree that always has height O(log N)-- instead
4 children. Red nodes represent the extra keys in 3-nodes and 4-nodes! leaf. Well, in worst case (the tree a binary tree! a! the parent now has few items, it needs fewer children, so we merge Red-Black vs. 2-3-4 Nodes! So lookup, insert, and delete will always be logarithmic in the number
to the right leaf node, and stick it in there. We established that binary search trees give us good run-times if For step 3 for inserting into a non-empty tree, what we need to do will depend on the color of K's parent. trees, and most commonly red-black trees. maintaining T's red-black tree properties. just pull it out, and voila! Balanced search trees have a height that is always O(log N). We know that all our operations are guaranteed to Sometimes we can fix that by taking an item from the parent and In order to handle this double-red situation, we will need to consider the color of G's other child, that is, P's sibling, S. (Note that S might be null, i.e., G only has one child and that child is P.) We have two cases: If P's sibling S is black or null, then we will do a trinode restructuring of K (the newly added node), P (K's parent), and G (K's grandparent). A Red-Black tree is a binary search tree in which each node is colored red or black! … The numbers mean a tree where every node with children (internal node) has either two, three, or four child nodes: An example of adding several values to a red-black tree will be presented in lecture. A red-black tree is a representation of a (2,4) tree by means of a binary tree whose nodes are colored redor black In comparison with its associated (2,4) tree, a red-black tree has Why else would we do it? Recoloring does not affect the black property of a tree: the number of black nodes on any path that goes through P and G is unchanged when P and G switch colors (similarly for S and G). An equivalent data structure of 2-3-4 trees is called a Red-Black tree. CS 16: Balanced Trees erm 218 Insertion into Red-Black Trees 1.Perform a standard search to find the leaf where the key should be added 2.Replace the leaf with an internal node with the new key 3.Color the incoming edge of the new node red 4.Add two new leaves, and color their incoming edges black 5.If the parent had an incoming red edge, we