Red-Black Trees

Question

I\'ve seen binary trees and binary searching mentioned in several books I\'ve read lately, but as I\'m still at the beginning of my studies in Computer Science, I\'ve yet to

Answer 1

Since you ask which tree people use, you need to know that a Red Black tree is fundamentally a 2-3-4 B-tree (i.e a B-tree of order 4). A B-tree is not equivalent to a binary tree(as asked in your question).

Here's an excellent resource describing the initial abstraction known as the symmetric binary B-tree that later evolved into the RBTree. You would need a good grasp on B-trees before it makes sense. To summarize: a 'red' link on a Red Black tree is a way to represent nodes that are part of a B-tree node (values within a key range), whereas 'black' links are nodes that are connected vertically in a B-tree.

So, here's what you get when you translate the rules of a Red Black tree in terms of a B-tree (I'm using the format Red Black tree rule => B Tree equivalent):

1) A node is either red or black. => A node in a b-tree can either be part of a node, or as a node in a new level.

2) The root is black. (This rule is sometimes omitted, since it doesn't affect analysis) => The root node can be thought of either as a part of an internal root node as a child of an imaginary parent node.

3) All leaves (NIL) are black. (All leaves are same color as the root.) => Since one way of representing a RB tree is by omitting the leaves, we can rule this out.

4)Both children of every red node are black. => The children of an internal node in a B-tree always lie on another level.

5)Every simple path from a given node to any of its descendant leaves contains the same number of black nodes. => A B-tree is kept balanced as it requires that all leaf nodes are at the same depth (Hence the height of a B-tree node is represented by the number of black links from the root to the leaf of a Red Black tree)

Also, there's a simpler 'non-standard' implementation by Robert Sedgewick here: (He's the author of the book Algorithms along with Wayne)

Answer 2

I'd like to address only the question "So what makes binary trees useful in some of the common tasks you find yourself doing while programming?"

This is a big topic that many people disagree on. Some say that the algorithms taught in a CS degree such as binary search trees and directed graphs are not used in day-to-day programming and are therefore irrelevant. Others disagree, saying that these algorithms and data structures are the foundation for all of our programming and it is essential to understand them, even if you never have to write one for yourself. This filters into conversations about good interviewing and hiring practices. For example, Steve Yegge has an article on interviewing at Google that addresses this question. Remember this debate; experienced people disagree.

In typical business programming you may not need to create binary trees or even trees very often at all. However, you will use many classes which internally operate using trees. Many of the core organization classes in every language use trees and hashes to store and access data.

If you are involved in more high-performance endeavors or situations that are somewhat outside the norm of business programming, you will find trees to be an immediate friend. As another poster said, trees are core data structures for databases and indexes of all kinds. They are useful in data mining and visualization, advanced graphics (2d and 3d), and a host of other computational problems.

I have used binary trees in the form of BSP (binary space partitioning) trees in 3d graphics. I am currently looking at trees again to sort large amounts of geocoded data and other data for information visualization in Flash/Flex applications. Whenever you are pushing the boundary of the hardware or you want to run on lower hardware specifications, understanding and selecting the best algorithm can make the difference between failure and success.

Answer 3

IME, almost no one understands the RB tree algorithm. People can repeat the rules back to you, but they don't understand why those rules and where they come from. I am no exception :-)

For this reason, I prefer the AVL algorithm, because it's easy to comprehend. Once you understand it, you can then code it up from scratch, because it make sense to you.

Answer 4

BSTs make the world go round, as said by Micheal. If you're looking for a good tree to implement, take a look at AVL trees (Wikipedia). They have a balancing condition, so they are guaranteed to be O(logn). This kind of searching efficiency makes it logical to put into any kind of indexing process. The only thing that would be more efficient would be a hashing function, but those get ugly quick, fast, and in a hurry. Also, you run into the Birthday Paradox (also known as the pigeon-hole problem).

What textbook are you using? We used Data Structures and Analysis in Java by Mark Allen Weiss. I actually have it open in my lap as i'm typing this. It has a great section about Red-Black trees, and even includes the code necessary to implement all the trees it talks about.

Answer 5

Trees can be fast. If you have a million nodes in a balanced binary tree, it takes twenty comparisons on average to find any one item. If you have a million nodes in a linked list, it takes five hundred thousands comparisons on average to find the same item.

If the tree is unbalanced, though, it can be just as slow as a list, and also take more memory to store. Imagine a tree where most nodes have a right child, but no left child; it is a list, but you still have to hold memory space to put in the left node if one shows up.

Anyways, the AVL tree was the first balanced binary tree algorithm, and the Wikipedia article on it is pretty clear. The Wikipedia article on red-black trees is clear as mud, honestly.

Beyond binary trees, B-Trees are trees where each node can have many values. B-Tree is not a binary tree, just happens to be the name of it. They're really useful for utilizing memory efficiently; each node of the tree can be sized to fit in one block of memory, so that you're not (slowly) going and finding tons of different things in memory that was paged to disk. Here's a phenomenal example of the B-Tree.

Answer 6

None of the answers mention what it is exactly BSTs are good for.

If what you want to do is just lookup by values then a hashtable is much faster, O(1) insert and lookup (amortized best case).

A BST will be O(log N) lookup where N is the number of nodes in the tree, inserts are also O(log N).

RB and AVL trees are important like another answer mentioned because of this property, if a plain BST is created with in-order values then the tree will be as high as the number of values inserted, this is bad for lookup performance.

The difference between RB and AVL trees are in the the rotations required to rebalance after an insert or delete, AVL trees are O(log N) for rebalances while RB trees are O(1). An example of benefit of this constant complexity is in a case where you might be keeping a persistent data source, if you need to track changes to roll-back you would have to track O(log N) possible changes with an AVL tree.

Why would you be willing to pay for the cost of a tree over a hash table? ORDER! Hash tables have no order, BSTs on the other hand are always naturally ordered by virtue of their structure. So if you find yourself throwing a bunch of data in an array or other container and then sorting it later, a BST may be a better solution.

The tree's order property gives you a number of ordered iteration capabilities, in-order, depth-first, breadth-first, pre-order, post-order. These iteration algorithms are useful in different circumstances if you want to look them up.

Red black trees are used internally in almost every ordered container of language libraries, C++ Set and Map, .NET SortedDictionary, Java TreeSet, etc...

So trees are very useful, and you may use them quite often without even knowing it. You most likely will never need to write one yourself, though I would highly recommend it as an interesting programming exercise.