Not sure if the question should be here or on programmers (or some other SE site), but I was curious about the relevant differences between balanced binary trees and indexable skiplists. The issue came up in the context of this question. From the wikipedia:
Skip lists are a probabilistic data structure that seem likely to supplant balanced trees as the implementation method of choice for many applications. Skip list algorithms have the same asymptotic expected time bounds as balanced trees and are simpler, faster and use less space.
Don't the space requirements of a skiplist depend on the depth of the hierarchy? And aren't binary trees easier to use, at least for searching (granted, insertion and deletion in balanced BSTs can be tricky)? Are there other advantages/disadvantages to skiplists?
(Some parts of your question (ease of use, simplicity, etc.) are a bit subjective and I'll answer them at the end of this post.)
Let's look at space usage. First, let's suppose that you have a binary search tree with n nodes. What's the total space usage required? Well, each node stores some data plus two pointers. You might also need some amount of information to maintain balance information. This means that the total space usage is
n * (2 * sizeof(pointer) + sizeof(data) + sizeof(balance information))
So let's think about an equivalent skiplist. You are absolutely right that the real amount of memory used by a skiplist depends on the heights of the nodes, but we can talk about the expected amount of space used by a skiplist. Typically, you pick the height of a node in a skiplist by starting at 1, then repeatedly flipping a fair coin, incrementing the height as long as you flip heads and stopping as soon as you flip tails. Given this setup, what is the expected number of pointers inside a skiplist?
An interesting result from probability theory is that if you have a series of independent events with probability p, you need approximately 1 / p trials (on expectation) before that event will occur. In our coin-flipping example, we're flipping a coin until it comes up tails, and since the coin is a fair coin (comes up heads with probability 50%), the expected number of trials necessary before we flip tails is 2. Since that last flip ends the growth, the expected number of times a node grows in a skiplist is 1. Therefore, on expectation, we would expect an average node to have only two pointers in it - one initial pointer and one added pointer. This means that the expected total space usage is
n * (2 * sizeof(pointer) + sizeof(data))
Compare this to the size of a node in a balanced binary search tree. If there is a nonzero amount of space required to store balance information, the skiplist will indeed use (on expectation) less memory than the balanced BST. Note that many types of balanced BSTs (e.g. treaps) require a lot of balance information, while others (red/black trees, AVL trees) have balance information but can hide that information in the low-order bits of its pointers, while others (splay trees) don't have any balance information at all. Therefore, this isn't a guaranteed win, but in many cases it will use space.
As to your other questions about simplicity, ease, etc: that really depends. I personally find the code to look up an element in a BST far easier than the code to do lookups in a skiplist. However, the rotation logic in balanced BSTs is often substantially more complicated than the insertion/deletion logic in a skiplist; try seeing if you can rattle off all possible rotation cases in a red/black tree without consulting a reference, or see if you can remember all the zig/zag versus zag/zag cases from a splay tree. In that sense, it can be a bit easier to memorize the logic for inserting or deleting from a skiplist.
Hope this helps!
And aren't binary trees easier to use, at least for searching
(granted, insertion and deletion in balanced BSTs can be tricky)?
Trees are "more recursive" (trees and subtrees) and SkipLists are "more iterative" (levels in an array). Of course, it depends on implementation, but SkipLists can also be very useful for practical applications.
It's easier to search in trees because you don't have to iterate levels in an array.
Are there other advantages/disadvantages to skiplists?
SkipLists are "easier" to implement. This is a little relative, but it's easier to implement a full-functional SkipList than deletion and balance operations in a BinaryTree.
Trees can be persistent (better for functional programming).
It's easier to delete items from SkipLists than internal nodes in a binary tree.
It's easier to add items to binary trees (keeping the balance is another issue)
Binary Trees are deterministic, so it's easier to study and analyze them.
My tip: If you have time, you must use a Balanced Binary Tree. If you have little time, use a Skip List. If you have no time, use a Library.
Something not mentioned so far is that skip lists can be advantageous for concurrent operations. If you read the source of ConcurrentSkipListMap, authored by Doug Lea... dig into the comments. It mentions:
there are no known efficient lock-free insertion and deletion algorithms for search trees. The immutability of the "down" links of index nodes (as opposed to mutable "left" fields in true trees) makes this tractable using only CAS operations.
You're right that this isn't the perfect forum.
The comment you quoted was written by the author of the original skip list paper: not exactly an unbiased assertion. It's been 23 years, and red-black trees still seem to be more prevalent than skip lists. An exception is redis key-value pair database, which includes skip lists as one option among its data structures.
Skip lists are very cool. But the only space advantage I've been able to show in the general randomized case is no need to store balance flags: two bits per value. This is assuming the hierarchy is dense enough to replicate binary tree performance. You can chalk this up as the price of determinism (vice. randomization). A nice feature of SL's is you can use less dense hierarchies to trade constant factors of speed for space.
Side note: it's not often discussed that if you don't need to traverse in sorted order, you can randomize unbalanced binary trees by just enciphering the keys (i.e. mapping to a pseudo-random cipher text with something very simple like RC4). Such trees are absolutely trivial to implement.
I've seen this data structure talked about a lot, but I am unclear as to what sort of problem would demand such a data structure (over alternative representations). I've never needed one, but perhaps that's because I don't quite grok it. Can you enlighten me?
One example of where you would use a binary search tree would be a sorted list of values where you want to be able to quickly add elements.
Consider using an array for this purpose. You have very fast access to read random values, but if you want to add a new value, you have to find the place in the array where it belongs, shift everything over, and then insert the new value.
With a binary search tree, you simply traverse the tree looking for where the value would be if it were in the tree already, and then add it there.
Also, consider if you want to find out if your sorted array contains a particular value. You have to start at one end of the array and compare the value you're looking for to each individual value until you either find the value in the array, or pass the point where it would have been. With a binary search tree, you greatly reduce the number of comparisons you are likely to have to make. Just a quick caveat, however, it is definitely possible to contrive situations where the binary search tree requires more comparisons, but these are the exception, not the rule.
One thing I've used it for in the past is Huffman decoding (or any variable-bit-length scheme).
If you maintain your binary tree with the characters at the leaves, each incoming bit decides whether you move to the left or right node.
When you reach a leaf node, you have your decoded character and you can start on the next one.
For example, consider the following tree:
.
/ \
. C
/ \
A B
This would be a tree for a file where the predominant letter was C (by having less bits used for common letters, the file is shorter than it would be for a fixed-bit-length scheme). The codes for the individual letters are:
A: 00 (left, left).
B: 01 (left, right).
C: 1 (right).
The class of problems you use then for are those where you want to be able to both insert and access elements reasonably efficiently. As well as unbalanced trees (such as the Huffman example above), you can also use balanced trees which make the insertions a little more costly (since you may have to rebalance on the fly) but make lookups a lot more efficient since you're traversing the minimum possible number of nodes.
from wiki
Self-balancing binary search trees can be used in a natural way to construct and maintain ordered lists, such as priority queues. They can also be used for associative arrays; key-value pairs are simply inserted with an ordering based on the key alone. In this capacity, self-balancing BSTs have a number of advantages and disadvantages over their main competitor, hash tables. One advantage of self-balancing BSTs is that they allow fast (indeed, asymptotically optimal) enumeration of the items in key order, which hash tables do not provide. One disadvantage is that their lookup algorithms get more complicated when there may be multiple items with the same key.
Self-balancing BSTs can be used to implement any algorithm that requires mutable ordered lists, to achieve optimal worst-case asymptotic performance. For example, if binary tree sort is implemented with a self-balanced BST, we have a very simple-to-describe yet asymptotically optimal O(n log n) sorting algorithm. Similarly, many algorithms in computational geometry exploit variations on self-balancing BSTs to solve problems such as the line segment intersection problem and the point location problem efficiently. (For average-case performance, however, self-balanced BSTs may be less efficient than other solutions. Binary tree sort, in particular, is likely to be slower than mergesort or quicksort, because of the tree-balancing overhead as well as cache access patterns.)
Self-balancing BSTs are flexible data structures, in that it's easy to extend them to efficiently record additional information or perform new operations. For example, one can record the number of nodes in each subtree having a certain property, allowing one to count the number of nodes in a certain key range with that property in O(log n) time. These extensions can be used, for example, to optimize database queries or other list-processing algorithms.
I want to save my objects according to a key in the attributes of my object in a sorted fashion. Later on I'll access these objects sequentially from max key to min key. I'll do some search tasks as well.
I consider to use either AVL tree or RB Tree. As far as i know they are nearly equivalent in theory(Both have O(logn)). But in practice which one might be better in performance in my situation. And is there a better alternative than those, considering that I'll mostly do insert and sequentially access to the ds.
Edit: I'm going to use java
For what it's worth, in C#, SortedDictionary<K, V> is implemented as a red-black tree, and in many implementations of the STL in C++, std::map<K, T> is implemented as a red-black tree.
Also, from Wikipedia on AVL vs. 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. This makes it
attractive for data structures that
may be built once and loaded without
reconstruction, such as language
dictionaries (or program dictionaries,
such as the order codes of an
assembler or interpreter).
Which ever is easiest for you to implement, you won't get better insertion than log(n) with a sorted list and we'd probably need a lot more detail than what you've provided to decide if there are other factors that make another structure more appropriate.
As you're doing it in Java, consider using a TreeSet (although it's a Set, so you can't have duplicate entries)...
Ok, so this is something that's always bothered me. The tree data structures I know of are:
Unbalanced binary trees
AVL trees
Red-black trees
2-3 trees
B-trees
B*-trees
Tries
Heaps
How do I determine what kind of tree is the best tool for the job? Obviously heaps are canonically used to form priority queues. But the rest of them just seem to be different ways of doing the same thing. Is there any way to choose the best one for the job?
Let’s pick them off one by one, shall we?
Unbalanced binary trees
For search tasks, never. Basically, their performance characteristics will be completely unpredictable and the overhead of balancing a tree won’t be so big as to make unbalanced trees a viable alternative.
Apart from that, unbalanced binary trees of course have other uses, but not as search trees.
AVL trees
They are easy to develop but their performance is generally surpassed by other balancing strategies because balancing them is comparatively time-intensive. Wikipedia claims that they perform better in lookup-intensive scenarios because their height is slightly less in the worst case.
Red-black trees
These are used inside most of C++’ std::map implemenations and probably in a few other standard libraries as well. However, there’s good evidence that they are actually worse than B(+) trees in every scenario due to caching behaviour of modern CPUs. Historically, when caching wasn’t as important (or as good), they surpassed B trees when used in main memory.
2-3 trees
B-trees
B*-trees
These require the most careful consideration of all the trees, since the different constants used are basically “magical” constans which relate in weird and sometimes unpredictable way to the underlying hardware architecture. For example, the optimal number of child nodes per level can depend on the size of a memory page or cache line.
I know of no good, general rule to distinguish between them.
Tries
Completely different. Tries are also search trees, but for text retrieval of substrings in a corpus. A trie is an uncompressed prefix tree (i.e. a tree in which the paths from root to leaf nodes correspond to all the prefixes of a given string).
Tries should be compared to, and offset against, suffix trees, suffix arrays and q-gram indices – not so much against other search trees because the data that they search is different: instead of discrete words in a corpus, the latter index structures allow a factor search.
Heaps
As you’ve already said, they are not search trees at all.
The same as any other data structure, you have to know the characteristics (complexity of search, insert, and delete operations) of each type of tree, and the requirements of the job you're selecting a tool for. The tree that has the best performance for the type of operations you'll do most often is usually the best tool for the job.
You can usually find the general characteristics for any kind of data structure on Wikipedia. Introduction to Algorithms also has at least a section (in some cases a whole chapter) on most of the data structures you've listed, so it's another good reference.
Similar question: When to choose RB tree, B-Tree or AVL tree?
Offhand, I'd say, write the simplest code that could possibly work (availing yourself of library-provided data structures if possible). Then measure its performance problems, if any.
If your performance needs are really extreme, read Konrad Rudolph's awesome answer. :)
Each of these has different complexity for insertion, deletion and retrieval, All have mostly O log(n) access times.
Each tree has specific characteristics which make them usefull in a certain way. You should compare there characteristics with the needs you have.
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 take a class that's really dealt with algorithms and data structures in a serious way.
I've checked around the typical sources (Wikipedia, Google) and most descriptions of the usefulness and implementation of (in particular) Red-Black trees have come off as dense and difficult to understand. I'm sure for someone with the necessary background, it makes perfect sense, but at the moment it reads like a foreign language almost.
So what makes binary trees useful in some of the common tasks you find yourself doing while programming? Beyond that, which trees do you prefer to use (please include a sample implementation) and why?
Red Black trees are good for creating well-balanced trees. The major problem with binary search trees is that you can make them unbalanced very easily. Imagine your first number is a 15. Then all the numbers after that are increasingly smaller than 15. You'll have a tree that is very heavy on the left side and has nothing on the right side.
Red Black trees solve that by forcing your tree to be balanced whenever you insert or delete. It accomplishes this through a series of rotations between ancestor nodes and child nodes. The algorithm is actually pretty straightforward, although it is a bit long. I'd suggest picking up the CLRS (Cormen, Lieserson, Rivest and Stein) textbook, "Introduction to Algorithms" and reading up on RB Trees.
The implementation is also not really so short so it's probably not really best to include it here. Nevertheless, trees are used extensively for high performance apps that need access to lots of data. They provide a very efficient way of finding nodes, with a relatively small overhead of insertion/deletion. Again, I'd suggest looking at CLRS to read up on how they're used.
While BSTs may not be used explicitly - one example of the use of trees in general are in almost every single modern RDBMS. Similarly, your file system is almost certainly represented as some sort of tree structure, and files are likewise indexed that way. Trees power Google. Trees power just about every website on the internet.
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.
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.
Red Black Trees and B-trees are used in all sorts of persistent storage; because the trees are balanced the performance of breadth and depth traversals are mitigated.
Nearly all modern database systems use trees for data storage.
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.
Red-black trees stay balanced, so you don't have to traverse deep to get items out. The time saved makes RB trees O(log()n)) in the WORST case, whereas unlucky binary trees can get into a lop sided configuration and cause retrievals in O(n) a bad case. This does happen in practice or on random data. So if you need time critical code (database retrievals, network server etc.) you use RB trees to support ordered or unordered lists/sets .
But RBTrees are for noobs! If you are doing AI and you need to perform a search, you find you fork the state information alot. You can use a persistent red-black to fork new states in O(log(n)). A persistent red black tree keeps a copy of the tree before and after a morphological operation (insert/delete), but without copying the entire tree (normally and O(log(n)) operation). I have open sourced a persistent red-black tree for java
http://edinburghhacklab.com/2011/07/a-java-implementation-of-persistent-red-black-trees-open-sourced/
The best description of red-black trees I have seen is the one in Cormen, Leisersen and Rivest's 'Introduction to Algorithms'. I could even understand it enough to partially implement one (insertion only). There are also quite a few applets such as This One on various web pages that animate the process and allow you to watch and step through a graphical representation of the algorithm building a tree structure.
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)
Lots and lots of heat here, but not much light, so lets see if we can provide some.
First, a RB tree is an associative data structure, unlike, say an array, which cannot take a key and return an associated value, well, unless that's an integer "key" in a 0% sparse index of contiguous integers. An array cannot grow in size either (yes, I know about realloc() too, but under the covers that requires a new array and then a memcpy()), so if you have either of these requirements, an array won't do. An array's memory efficiency is perfect. Zero waste, but not very smart, or flexible - realloc() not withstanding.
Second, in contrast to a bsearch() on an array of elements, which IS an associative data structure, a RB tree can grow (AND shrink) itself in size dynamically. The bsearch() works fine for indexing a data structure of a known size, which will remain that size. So if you don't know the size of your data in advance, or new elements need to be added, or deleted, a bsearch() is out. Bsearch() and qsort() are both well supported in classic C, and have good memory efficiency, but are not dynamic enough for many applications. They are my personal favorite though because they're quick, easy, and if you're not dealing with real-time apps, quite often are flexible enough. In addition, in C/C++ you can sort an array of pointers to data records, pointing to the struc{} member, for example, you wish to compare, and then rearranging the pointer in the pointer array such that reading the pointers in order at the end of the pointer sort yields your data in sorted order. Using this with memory-mapped data files is extremely memory efficient, fast, and fairly easy. All you need to do is add a few "*"s to your compare function/s.
Third, in contrast to a hashtable, which also must be a fixed size, and cannot be grown once filled, a RB tree will automagically grow itself and balance itself to maintain its O(log(n)) performance guarantee. Especially if the RB tree's key is an int, it can be faster than a hash, because even though a hashtable's complexity is O(1), that 1 can be a very expensive hash calculation. A tree's multiple 1-clock integer compares often outperform 100-clock+ hash calculations, to say nothing of rehashing, and malloc()ing space for hash collisions and rehashes. Finally, if you want ISAM access, as well as key access to your data, a hash is ruled out, as there is no ordering of the data inherent in the hashtable, in contrast to the natural ordering of data in any tree implementation. The classic use for a hash table is to provide keyed access to a table of reserved words for a compiler. It's memory efficiency is excellent.
Fourth, and very low on any list, is the linked, or doubly-linked list, which, in contrast to an array, naturally supports element insertions and deletions, and as that implies, resizing. It's the slowest of all the data structures, as each element only knows how to get to the next element, so you have to search, on average, (element_knt/2) links to find your datum. It is mostly used where insertions and deletions somewhere in the middle of the list are common, and especially, where the list is circular and feeds an expensive process which makes the time to read the links relatively small. My general RX is to use an arbitrarily large array instead of a linked list if your only requirement is that it be able to increase in size. If you run out of size with an array, you can realloc() a larger array. The STL does this for you "under the covers" when you use a vector. Crude, but potentially 1,000s of times faster if you don't need insertions, deletions or keyed lookups. It's memory efficiency is poor, especially for doubly-linked lists. In fact, a doubly-linked list, requiring two pointers, is exactly as memory inefficient as a red-black tree while having NONE of its appealing fast, ordered retrieval characteristics.
Fifth, trees support many additional operations on their sorted data than any other data structure. For example, many database queries make use of the fact that a range of leaf values can be easily specified by specifying their common parent, and then focusing subsequent processing on the part of the tree that parent "owns". The potential for multi-threading offered by this approach should be obvious, as only a small region of the tree needs to be locked - namely, only the nodes the parent owns, and the parent itself.
In short, trees are the Cadillac of data structures. You pay a high price in terms of memory used, but you get a completely self-maintaining data structure. This is why, as pointed out in other replies here, transaction databases use trees almost exclusively.
If you would like to see how a Red-Black tree is supposed to look graphically, I have coded an implementation of a Red-Black tree that you can download here
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.
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.