ICS 311 #11: Balanced Trees (2-4 and Red-Black)


Outline

Readings and Screencasts

The presentation of Red-Black Trees (RBTs) in Cormen et al. omits a major conceptual motivation: RBTs are a binary representation of a balanced multi-way tree, and the complex operations on RBTs correspond to simpler operations on 2-4 trees. This motivation is nicely presented by Sedgewick, so read in this order:

Although it is a good start, Sedgewick's presentation of 2-4 trees omits some detail, so I draw on Goodrich & Tamassia's slides widely available on the Web for the following presentation.


Multi-Way Trees

A multi-way search tree is an ordered tree such that

Multi-way inorder traversal can be defined by extension of BST inorder traversal to visit the keys in increasing order:

Visit item (ki, oi) of node v between the recursive traversals of the subtrees of v rooted at children vi and vi+1.

Searching can similarly be extended to multi-way trees by searching within each node as well as down the tree:

For example, searching for key 30, we follow the red path:


2-4 Trees

Also known as (2,4) or 2-3-4 trees, these are multi-way trees restricted in two ways:

The internal nodes are called 2-nodes, 3-nodes or 4-nodes, depending on the number of children they have.

Height of 2-4 Trees and Searching

Theorem: A 2-4 tree storing n items has height Θ(lg n).

Let h be the height of a 2-4 tree with n items. The proof is in two parts, for the upper bound and the lower bound.

Upper bound: The tallest possible tree for a fixed n is when all internal nodes are 2-nodes, each storing 1 item; i.e., the tree is equivalent to a binary tree. By the depth property, it is a complete binary tree. (See also quantitative facts for complete binary trees in Topic 8, but we do the full proof here.)

In such a tree, there are 1=1*20 items stored at depth 0, 2=1*21 items stored at depth 1, 4=1*22 items stored at depth 2, or more generally 1*2i items at depth i. (I include the "1*" to represent the number of items stored in a node, which will change in the second part of the proof, but drop it now.) Since the last level of a 2-4 tree of height h stores no items (see for example picture above) we stop at h−1. So the total number of nodes is Σi=0,h−12i.

Applying formula A5 (shown) with h−1 for n and 2 for x, the number n of nodes stored in the tallest 2-4 tree of height h is:

Σi=0,h−12i
            = (2(h−1)+1 − 1)/(2−1)
            = 2h − 1,

so
        n = 2h − 1   or   n + 1 ≥ 2h

Taking the log base 2 of both sides:   lg (n + 1) ≥ h.   Thus, h = O(lg n).

Lower bound: The shortest possible tree for a fixed n is when all internal nodes are 4-nodes, each storing 3 items (by the size property). By the depth property, all levels are filled out. The proof is similar to above but with different constants:

There are 3=3*40 items stored at depth 0, 12=3*41 items stored at depth 1, 48=3*42 items stored at depth 2, or more generally 3*4i items at depth i. Since the last level of a 2-4 tree of height h stores no items we stop at h−1. So the total number of nodes is Σi=0,h−13*4i. Moving the 3 out and applying formula A5,

3*Σi=0,h−14i
            = 3*(4(h−1)+1− 1)/(4−1)
            = 4h−1,
so
        n = 4h−1   or   n + 1 = 4h

Taking the log base 4 of both sides:   log4 (n + 1) = h.
Since the base of logarithms differs only by a constant factor (see 3.15 of text), in this case by log24 = 2, h = Ω(lg n).

Putting these results together, h = Θ(lg n).

Since searching in a 2-4 tree with n items requires time proportional to a path from root to leaves, searching requires O(lg n) time.

2-4 Tree Insertion

We will examine insertion and deletion briefly to understand the conceptual cases.

Insert a new item keyed by k into (not below) the parent of the leaf reached by searching for k. (In this respect, 2-4 trees differ from binary search trees.)

This preserves depth but may cause overflow (a node may become a 5-node).

Example: Inserting 30, we find its position between 27 and 32. However inserting here causes overflow:

Overflow is handled with a split operation, as illustrated below with a simpler tree:

(Note: Sedgewick splits 4-nodes on the way down while searching for the insertion position, guaranteeing that there will be no overflow. Both Goodrich & Tamassia and Cormen et al. take the other approach, propagating splits upwards only as needed. The asymptotic time complexity remains the same.)

Time Complexity of 2-4 Insertion

A tree with n items has Θ(lg n) height. The algorithm first searches for the insertion location, which may require visiting h = Θ(lg n) nodes (Θ, not O, because we must go to the leaves in all cases). The insertion takes Θ(1) time. If there is overflow, splits (taking Θ(1) time each) may be propagated upwards to as many as O(lg n) nodes. Since the Θ(lg n) overrides the possibility of slower growing functions in O(lg n), insertion is Θ(lg n).

2-4 Tree Deletion

If the entry to be deleted is in a node that has internal nodes as children, we replace the entry to be deleted with its inorder successor and delete the latter entry. Example: to delete key 24, we replace it with 27 (inorder successor):

This reduces deletion of an entry to the case where the item is at the node with leaf children.

Deletion of an entry from a node v may cause underflow, where node v becomes a 1-node with one child and no keys. Underflow at node v with parent u is handled in two cases.

Case 1: An adjacent sibling of v is a 2-node. Perform a fusion operation, merging v with the adjacent 2-node sibling w and moving a key from u to the merged node v'.

After a fusion, the underflow may propagate to the parent u, for at most O(lg n) adjustments up the tree.

Case 2: An adjacent sibling w of v is a 3-node or a 4-node. Perform a transfer operation: move a key of w to v; a key from u to v; and a key from w to u:

The underflow is eliminated. (Which keys are moved depends on the configuration. In this example we move the largest key out of each node, but it could be different if underflow was in a non-rightmost child or it was a right sibling that was a 3-node or 4-node.)

Time Complexity of 2-4 Deletion

The algorithm first searches for the item to delete, which requires visiting h = Θ(lg n) nodes on the way down the tree, either to find a bottom level key to delete, or to find the successor of a key in an internal node to delete. Underflow is handled with up to O(lg n) fusions and transfers, each taking Θ(1) time. Thus deletion is Θ(lg n).


Red-Black Trees

Red-Black Tree Properties

A red-black tree (RBT) is a binary search tree with the following additional properties:

  1. Color property: Every node is either red or black. (We can indicate this either by coloring the node or by coloring its parent link.)
  2. Root property: The root is black
  3. External property: Every leaf is black.
  4. Internal property: If a node is red, then both of its children are black. (Hence, no two reds in a row are allowed on a simple path from the root to a leaf.)
  5. Depth property: For each node, all the paths from the node to descendant leaves contain the same number of black nodes (the black height of the node).

These properties seem rather arbitrary until we consider the correspondence with 2-4 trees shortly, but first please verify how the properties hold in this example ...

Red-Black Tree Representation

A single extra bit is required on each node to mark it as "red" or "black".

To save space, we can represent the leaf nodes and the parent of the root node with a single node, T.nil:

This also simplifies the code, as we can follow pointers without having to check for null pointers.

We usually don't draw T.nil:

RBTs as a Binary Representation of 2-4 Trees

It would be rather complex to implement and manipulate 2-nodes, 3-nodes and 4-nodes. One motivation for red-black trees is that they provide a binary tree representation of 2-4 trees, enabling us to manipulate only one kind of node. The mapping is as follows (you should make sure you understand this well before going on!):

Red nodes (and the links from their parents) capture the internal structure of a (2,3) node;

Black nodes (and the links from their parents) capture the structure of the (2,3) tree itself.

RBTs as Binary Search Trees

At the same time as they represent 2-4 trees, RBTs are also Binary Search Trees: they satisfy the Binary Search Tree property. For example, here is a RBT: we can search for keys or enumerate elements in order as usual:

In order to maintain the Red-Black-Tree properties, it will be necessary to do structural rotations. These rotations are designed to not disrupt the BST property. For example, this rotation does not disturb the BST ordering of keys 9, 11, 12, 14, 17, 18, 19:

Height of Red-Black Trees and Searching

Theorem: A red-black tree storing n items has height Θ(lg n).

Proof:

(See Cormen et al. for a proof not relying on 2-4 trees.)

Therefore, searching in a red-black tree with n items takes O(lg n) time (O rather than Θ as we may find the key in an internal node).

We now consider insertion and deletion. Please see the CLRS textbook for the many details of implementation in pseudocode, etc.: here we will concentrate on seeing how the RBT operations correspond to 2-4 tree operations.

Insertion in Red-Black Trees

To insert an element with key k, perform the insertion for binary search trees (except that conceptually we insert k in an internal node with null children, not at a leaf node), and color the newly inserted node z red, unless it is the root.

This preserves the color, root, external, and depth properties. (You should check this in the example below.)

If the parent v of z is black, this also preserves the internal property and we are done.

Else (v is red), we have a double red (i.e., a violation of the internal property), which requires a reorganization of the tree. For example, insert 4:

A double red with child z and parent v is dealt with in two cases. Let w be the sibling of v (and hence the uncle of z).

Case 1: w is black. The double red is an incorrect representation of a 4-node. (We will fix this with restructuring). For example, the RBT on the left is an incorrect representation of the 2-4 tree on the right:

Case 2: w is red. The double red corresponds to an overflow in the 2-4 tree. (We will fix this with recoloring, which is the equivalent of a 2-4 split.) For example:

Restructuring

Restructuring remedies a child-parent double red when the parent red node has a black sibling. It restores the correct representation (internal property) of a 4-node, leaving other RBT and BST properties intact:

There are four restructuring configurations depending on whether the double red nodes are left or right children. They all lead to the same end configuration of a black with two red children:

After a restructuring, the double red has been remedied without violating any of the other properties (you should verify this): there is no need to propagate changes upwards.

Notice that the height of the subtree tree has been reduced by one. This is the operation that keeps the trees balanced to within a constant factor of lg(n) height, by ensuring that the height of the RBT is no more than twice that of the (necessarily balanced) 2-4 tree it represents. Do you see why?

Recoloring

Recoloring remedies a child-parent double red when the parent red node has a red sibling. The parent v and its sibling w become black and the grandparent u becomes red, unless it is the root. We recolor a non-root grandparent in order to maintain equal black height (in case the grandparent has a sibling).

Recoloring is equivalent to performing a split on a 5-node in a 2-4 tree. (When there is a double red and yet another red in the parent's sibling, we are trying to collect too many keys under the grandparent.) For example, the RBT recoloring on the top corresponds to the 2-4 transformation on the bottom:

Notice that in this example the parent "4" is now red, meaning it belongs to its parent node in the 2-4 tree. The double red violation may propagate to this parent in the RBT, which corresponds to the overflow propagating up the 2-4 tree, requiring further repair.

Be aware that 3-nodes and overloaded nodes have more than one possible representation as a red-black tree, and in particular this affects which key is promoted by the recoloring. For example, the above overloaded 2-4 node of keys {2, 4, 6, 7} can also be represented by either of the following trees, and you can easily verify that recoloring will promote the key 6 in the 2-4 tree.

Time Complexity of RBT Insertion

We already established that insertion in 2-4 trees is Θ(lg n) due to their height. Since RBTs are only at most twice as high, we might expect this result to transfer, and it does, but it needs to be shown separately since the manipulations of the RBT are different. So:

Thus insertion is Θ(lg n).

Note: A top-down version of this algorithm is also possible, restructuring on the way down and requiring only one pass through the tree. See the Sedgewick reading distributed.

Deletion in Red-Black Trees

To remove item with key k, we first perform the BST deletion (modified for our representational changes using T.nil).

Because deletion of a node higher in the tree involves replacing it with its successor, which is then deleted, deletion always involves an internal and an external node.

We can preserve the RBT properties at the new internal location of the successor by giving the successor the color of the node deleted, so we need only be concerned with possible violations of RBT properties at the bottom of the tree, where the successor was moved from, or where a node without a successor was deleted.

Let v be the internal node removed, w the external node removed, and r the sibling of w:

    x       
     \               x                    
      v       ==>     \
     / \               r 
    r   w

If either v or r was red, we color r black and we are done (the number of black nodes has not changed).

Else (v and r were both black), we have removed a black node, violating the depth property. We "fix" this by coloring r double black, a fictional color. (Intuitively, the black of both v and r have been absorbed into r.) Now we have the correct "amount" of black on this path from root to leaf, but the double black violates the color property.

Fixing this will require a reorganization of the tree. Example: deletion of 8 causes a double black:

A double black corresponds to underflow in 2-4 trees (and here the images I am borrowing from Goodrich & Tamassia go to greyscale!):

Goodrich & Tamassia's algorithm for remedying a double black node r with sibling y considers three cases, discussed below. (Note that these are different from CLRS's four cases!)

Case 1: y is black and has a red child: Perform a RBT restructuring, equivalent to a 2-4 transfer, and we are done.

For example, if we have the RBT on the left corresponding to underflow in the 2-4 tree on the right:

... we do the following restructuring:

Case 2: y is black and its children are both black: Perform a RBT recoloring, equivalent to a 2-4 fusion, which may propagate up the double black violation.

If the double-black reaches the root we can just remove it, as it is now on all of the paths from the root to the leaves, so does not affect property 5, the depth property.

For example, if we have the RBT on the left corresponding to underflow in the 2-4 tree on the right:

... we do the following recoloring: the black node y is colored red, and the double black node r is colored ordinary black:

The root of the above subtree takes on an extra black, which propagates only if it was previously black and is not the root. If it was red it merely turns black; if it was the root the extra black no longer affects the balanced black height of the tree.

Case 3: y is red: Perform a RBT adjustment, equivalent to choosing a different representation of a 3-node, after which either Case 1 or Case 2 applies.

These are both representations of the following 2-4 tree, but the transformation allows one of the other cases to apply, reducing duplication of cases.

The CLRS chapter divides the situation up into four cases: try to see whether you can map between the above cases and theirs!

Time Complexity of RBT Deletion

The analysis is similar to the previous ones: Θ(lg n) search to find the deletion point (the item to delete may be in an internal node, but we always find its successor in any case, which is at the bottom of the tree), followed by deletion and restructuring O(1) operations that are propagated at most up O(lg n) levels. Deletion is Θ(lg n).

RBT Animations

You may want to look at these:

http://secs.ceas.uc.edu/~franco/C321/html/RedBlack/redblack.html
A java applet. You can go step by step and it tells you the rules violated and the fixes. Must click on "next step" until done with process. To delete, click on Delete and then on the node to be deleted.
http://www.csanimated.com/animation.php?t=Red-black_tree
A flash animation: slides with voice-over. It goes kind of fast (little time to figure out what property is being fixed in each case), and does not let you control slide by slide.

Related Data Structures

AVL Trees, named for their authors, are the oldest balanced trees. They are binary trees with the requirement that the heights of the left and right subtree of any given node differ at most by 1. A small amount of extra storage is needed to record height differences. Their operations are O(lg n) like RBTs, but may require O(lg n) rotations to rebalance.

Splay Trees are binary trees in which an adjustment moving a node towards the root called splaying is done after every access (including search). There are no rules about properties to maintain and no labels. Amazingly, splaying alone is enough to guarantee O(lg n) behavior in an amortized sense: we will use these as an example when we cover chapter 17 Amortized analysis. They also make frequently accessed items more accessible.

B-Trees, covered in Chapter 18 of Cormen et al. (but not in this course), are balanced multi-way trees that allow up to M keys per node for large M. They are used for trees in external (disk) storage, where speed is optimized by making the size of a node be the same as the size of a block read in by one disk read.


Comparison of Dictionary Implementations

First, here is a summary of the correspondence between 2-4 and Red-Black tree operations:

Insertion: Remedy double red
2-4 tree action
Red-Black Tree Action
Result
Change of 4-node representation Restructuring Double red removed
Split Recoloring Double red removed or propagated up
 
Deletion: Remedy double black
2-4 tree action
Red-Black Tree Action Result
Transfer Restructuring Double black removed
Fusion Recoloring Double black removed or propagated up
Change of 3-node representation Adjustment Restructuring or recoloring follows

A comparison of run times.

  Search Insert Delete Notes
Hash Table O(1) expected O(1) expected O(1) expected

No ordered dictionary methods. Simple to implement.

Doubly Linked List O(n) O(1) if not sorted; O(n) if sorted Θ(1) if node given, O(n) otherwise Simple to implement.
Skip List O(lg n) with high probability O(lg n) with high probability O(lg n) with high probability Randomized insertion. Simple to implement.
Binary Tree O(n) worst case, O(lg n) random O(n) worst case, O(lg n) random O(n) worst case, O(lg n) random Moderately complex to implement deletion.
Red-Black Tree O(lg n) worst case Θ(lg n) Θ(lg n) Complex to implement.

From this we can see that hash tables are most efficient expected behavior when no ordered methods are needed, and red-black trees give us the best guarantee when ordering matters, althogh skip lists are a viable alternative if you have to implement it yourself and want simpler code. But there is good news! Java trees implement the CLRS algorithms for red-black trees!


Dan Suthers
Last modified: Sat Oct 7 03:47:58 HST 2017
Images are from lecture slides provided by Michael Goodrich and Roberto Tamassia, and from the instructor's material for Cormen et al. Introduction to Algorithms, Third Edition.