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A binary tree is made of nodes, where each node contains a "left" pointer, a "right" pointer, and a data element. The "root" pointer points to the topmost node in the tree. The left and right pointers recursively point to smaller "subtrees" on either side. A null pointer represents a binary tree with no elements -- the empty tree. The formal recursive definition is: a binary tree is either empty (represented by a null pointer), or is made of a single node, where the left and right pointers (recursive definition ahead) each point to a binary tree.

A binary search tree (BST) or "ordered binary tree" is a type of binary tree where the nodes are arranged in order: for each node, all elements in its left subtree are less to the node (<), and all the elements in its right subtree are greater than the node (>).

   / \
  3   6 
 / \   \
1   4   9

The tree shown above is a binary search tree -- the "root" node is a 5, and its left subtree nodes (1, 3, 4) are < 5, and its right subtree nodes (6, 9) are > 5. Recursively, each of the subtrees must also obey the binary search tree constraint: in the (1, 3, 4) subtree, the 3 is the root, the 1 < 3 and 4 > 3.

Watch out for the exact wording in the problems -- a "binary search tree" is different from a "binary tree".

What "tree shown above?" Did you forget to include a tree image in your answer?

Difference between binary tree and binary search tree - Stack Overflow

binary-tree binary-search-tree
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A binary search tree is a special kind of binary tree which exhibits the following property: for any node n, every descendant node's value in the left subtree of n is less than the value of n, and every descendant node's value in the right subtree is greater than the value of n.

Difference between binary tree and binary search tree - Stack Overflow

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Binary Search Tree (BST) on the other hand, is a special form of Binary Tree data structure where each node has a comparable value, and smaller valued children attached to left and larger valued children attached to the right.

Thus, all BST's are Binary Tree however only some Binary Tree's may be also BST. Notify that BST is a subset of Binary Tree.

So, Binary Tree is more of a general data-structure than Binary Search Tree. And also you have to notify that Binary Search Tree is a sorted tree whereas there is no such set of rules for generic Binary Tree.

Binary Tree
       /   \
      /     \
     9       2
    / \     / \
  15   17  19  21
       /    \
      /      \
     25      75
    /  \    /  \
  20    30 70   80

Also notify that for any parent node in the BST;

There is no such a rule for Binary Tree Node. The only rule for Binary Tree Node is having two childrens so it self-explains itself that why called binary.

Difference between binary tree and binary search tree - Stack Overflow

binary-tree binary-search-tree
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To squabble about the performance of binary-trees is meaningless - they are not a data structure, but a family of data structures, all with different performance characteristics. While it is true that unbalanced binary trees perform much worse than self-balancing binary trees for searching, there are many binary trees (such as binary tries) for which "balancing" has no meaning.

  • Binary Search Tree - Used in many search applications where data is constantly entering/leaving, such as the map and set objects in many languages' libraries.
  • Binary Space Partition - Used in almost every 3D video game to determine what objects need to be rendered.
  • Binary Tries - Used in almost every high-bandwidth router for storing router-tables.
  • Hash Trees - used in p2p programs and specialized image-signatures in which a hash needs to be verified, but the whole file is not available.
  • Heaps - Used in implementing efficient priority-queues, which in turn are used for scheduling processes in many operating systems, Quality-of-Service in routers, and A* (path-finding algorithm used in AI applications, including robotics and video games). Also used in heap-sort.
  • Huffman Coding Tree (Chip Uni) - used in compression algorithms, such as those used by the .jpeg and .mp3 file-formats.
  • GGM Trees - Used in cryptographic applications to generate a tree of pseudo-random numbers.
  • Treap - Randomized data structure used in wireless networking and memory allocation.
  • T-tree - Though most databases use some form of B-tree to store data on the drive, databases which keep all (most) their data in memory often use T-trees to do so.

The reason that binary trees are used more often than n-ary trees for searching is that n-ary trees are more complex, but usually provide no real speed advantage.

In a (balanced) binary tree with m nodes, moving from one level to the next requires one comparison, and there are log_2(m) levels, for a total of log_2(m) comparisons.

In contrast, an n-ary tree will require log_2(n) comparisons (using a binary search) to move to the next level. Since there are log_n(m) total levels, the search will require log_2(n)*log_n(m) = log_2(m) comparisons total. So, though n-ary trees are more complex, they provide no advantage in terms of total comparisons necessary.

> Treap - Randomized data structure used in wireless networking and memory allocation. How exactly they are used in memory allocation and wireless networking?

There are lots of useful data structures and algorithms that make use of the word "binary", and "binary SEARCH tree" is in fact one of them, but that is not the question that was asked. What use does a plain old "binary tree" have, not a sorted one, not a balanced one, not a full one. Just a plain old random tree?

@MichaelErickson Did you, uh, read the answer? Because that's exactly the question I answered.

Hastables are binary trees? Even though you meant that hashtables can be implemented using binary trees, that's rarely the case, I think. Note also that the original question was about binary trees and not general trees. Syntax trees are not necessarily binary.

What are the applications of binary trees? - Stack Overflow

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When most people talk about binary trees, they're more often than not thinking about binary search trees, so I'll cover that first.

That's because, unless the data is coming in in a relatively random order, the tree can easily degenerate into its worst-case form, which is a linked list, since simple binary trees are not balanced.

A good case in point: I once had to fix some software which loaded its data into a binary tree for manipulation and searching. It wrote the data out in sorted form:


so that, when reading it back in, ended up with the following tree:

 /     \
=       Bob
       /   \
      =     Chloe
           /     \
          =       David
                 /     \
                =       Edwina
                       /      \
                      =        Frank
                              /     \
                             =       =

which is the degenerate form. If you go looking for Frank in that tree, you'll have to search all six nodes before you find him.

Binary trees become truly useful when you balance them. This involves rotating sub-trees through their root node so that the height difference between any two sub-trees is less than or equal to 1. Adding those names above one at a time into a balanced tree would give you the following sequence:

1.   Alice
    /     \
   =       =
2.   Alice
    /     \
   =       Bob
          /   \
         =     =
3.        Bob
        _/   \_
   Alice       Chloe
  /     \     /     \
 =       =   =       =
4.        Bob
        _/   \_
   Alice       Chloe
  /     \     /     \
 =       =   =       David
                    /     \
                   =       =
5.           Bob
        ____/   \____
   Alice             David
  /     \           /     \
 =       =     Chloe       Edwina
              /     \     /      \
             =       =   =        =
6.              Chloe
            ___/     \___
         Bob             Edwina
        /   \           /      \
   Alice     =      David        Frank
  /     \          /     \      /     \
 =       =        =       =    =       =

You can actually see whole sub-trees rotating to the left as the entries are added and this gives you a balanced binary tree in which the worst case lookup is O(log N) rather than O(N) as the degenerate form gives. At no point does the highest NULL (=) differ from the lowest by more than one level. And, in the final tree above, you can find Frank by only looking at three nodes (Chloe, Edwina and, finally, Frank).

Of course, they become even more useful when you make them balanced multi-way trees rather than binary tress. That means that each node holds more than one item (technically, they hold N items and N+1 pointers, a binary tree being a special case of a 1-way multi-way tree with 1 item and 2 pointers).

Alice Bob Chloe
 /     |   |     \
=      =   =      David Edwina Frank
                 /     |      |     \
                =      =      =      =

This is typically used in maintaining keys for an index of items. I've written database software optimised for the hardware where a node is exactly the size of a disk block (say, 512 bytes) and you put as many keys as you can into a single node. The pointers in this case were actually record numbers into a fixed-length-record direct-access file separate from the index file (so record number X could be found by just seeking to X * record_length).

For example, if the pointers are 4 bytes and the key size is 10, the number of keys in a 512-byte node is 36. That's 36 keys (360 bytes) and 37 pointers (148 bytes) for a total of 508 bytes with 4 bytes wasted per node.

The use of multi-way keys introduces the complexity of a two-phase search (multi-way search to find the correct node combined with a small sequential search to find the correct key in the node) but the advantage in doing less disk I/O more than makes up for this.

I see no reason to do this for an in-memory structure, you'd be better off sticking with a balanced binary tree and keeping your code simple.

Also keep in mind that the advantages of O(log N) over O(N) don't really appear when your data sets are small. If you're using a multi-way tree to store the fifteen people in your address book, it's probably overkill. The advantages come when you're storing something like every order from your hundred thousand customers over the last ten years.

The whole point of big-O notation is to indicate what happens as the N approaches infinity. Some people may disagree but it's even okay to use bubble sort if you're sure the data sets will stay below a certain size, as long as nothing else is readily available :-)

As to other uses for binary trees, there are a great many, such as:

  • Binary heaps where higher keys are above or equal to lower ones rather than to the left of (or below or equal to and right);
  • Abstract syntax trees for compilation of computer languages;
  • Huffman trees for compression of data;

Given how much explanation I generated for the search trees, I'm reticent to go into a lot of detail on the others, but that should be enough to research them, should you desire.

+1 For such a we written answer; plus introducing me to balanced multi-way trees, something I've not come across before.

I disagree with your assertion that they are useful for little other than educating students. The are quite useful, even as a simple static data structure. However, this is a very well written and illustrated answer, so +1 for all the rest. :-)

On modern hardware, nearly all trees should be multi-way.

What are the applications of binary trees? - Stack Overflow

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 / \
2   3

Binary search tree: Used for searching. A binary tree where the left child contains only nodes with values less than the parent node, and where the right child only contains nodes with values greater than or equal to the parent.

 / \
1   3

What is the point of the "non-search" binary tree?

@pete: It's a conceptual thing, you won't necessarily ever actually make one that is completely unconstrained. However, there are lots of non-search binary trees that are special in some other way, e.g. binary heaps.

@pete birary trees do not necessarily have to contain comparable data, many (non search) binary trees are used for parsing algebraic expression, binary tree is perfect t write a infix notation parser, by placing the operator as node(s) and numerical values as leafs

@JBoy: They're not going to be binary trees in that case though. (e.g. unary operators can't have two children.) I really can't think of a practical use case for unconstrained binary trees, that's why I made that comment.

Difference between binary tree and binary search tree - Stack Overflow

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a binary tree is a whose children are not more than two while a binary search tree is a tree that follows the variant property which says that, the left child should be less than the root node's key and the right child should be greater than the root node's key.

Difference between binary tree and binary search tree - Stack Overflow

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The organization of Morse code is a binary tree.

What are the applications of binary trees? - Stack Overflow

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A binary tree is a tree data structure in which each node has at most two child nodes, usually distinguished as "left" and "right". Nodes with children are parent nodes, and child nodes may contain references to their parents. Outside the tree, there is often a reference to the "root" node (the ancestor of all nodes), if it exists. Any node in the data structure can be reached by starting at root node and repeatedly following references to either the left or right child. In a binary tree a degree of every node is maximum two.

Binary trees are useful, because as you can see in the picture, if you want to find any node in the tree, you only have to look a maximum of 6 times. If you wanted to search for node 24, for example, you would start at the root.

  • The root has a value of 31, which is greater than 24, so you go to the left node.
  • The left node has a value of 15, which is less than 24, so you go to the right node.
  • The right node has a value of 27, which is greater than 24, so you go to the left node.
  • The left node has a value of 25, which is greater than 24, so you go to the left node.
  • The node has a value of 24, which is the key we are looking for.

You can see that you can exclude half of the nodes of the entire tree on the first pass. and half of the left subtree on the second. This makes for very effective searches. If this was done on 4 billion elements, you would only have to search a maximum of 32 times. Therefore, the more elements contained in the tree, the more efficient your search can be.

Deletions can become complex. If the node has 0 or 1 child, then it's simply a matter of moving some pointers to exclude the one to be deleted. However, you can not easily delete a node with 2 children. So we take a short cut. Let's say we wanted to delete node 19.

Since trying to determine where to move the left and right pointers to is not easy, we find one to substitute it with. We go to the left sub-tree, and go as far right as we can go. This gives us the next greatest value of the node we want to delete.

Now we copy all of 18's contents, except for the left and right pointers, and delete the original 18 node.

To create these images, I implemented an AVL tree, a self balancing tree, so that at any point in time, the tree has at most one level of difference between the leaf nodes (nodes with no children). This keeps the tree from becoming skewed and maintains the maximum O(log n) search time, with the cost of a little more time required for insertions and deletions.

Here is a sample showing how my AVL tree has kept itself as compact and balanced as possible.

In a sorted array, lookups would still take O(log(n)), just like a tree, but random insertion and removal would take O(n) instead of the tree's O(log(n)). Some STL containers use these performance characteristics to their advantage so insertion and removal times take a maximum of O(log n), which is very fast. Some of these containers are map, multimap, set, and multiset.

You only have to search O(log n) if you're dealing with a binary search tree (that is itself well-balanced.) Arbitrary binary trees have no ordering constraints, and a random BST has search complexity O(log h).

Those are not the kind stored in the relevant Standard containers.

What are the applications of binary trees? - Stack Overflow

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Binary Tree is a specialized form of tree with two child (left child and right Child). It is simply representation of data in Tree structure

Binary Search Tree (BST) is a special type of Binary Tree that follows following condition:

  • left child node is smaller than its parent Node
  • right child node is greater than its parent Node

These conditions are not sufficient. The entire left subtree must contain no keys only less than the parent's, and the entire right subtree must contain nodes greater.

Difference between binary tree and binary search tree - Stack Overflow

binary-tree binary-search-tree
Rectangle 27 3

Assuming you mean "binary search tree" rather than "binary tree", you don't need any modifications to find the minimum element y in the tree such that y >= x.

search(n, x, best_so_far) ::=
    if n == nil { return best_so_far }
    if n.value == x { return x }
    if n.value > x { return search(n.left, x, min(best_so_far, n.value) }
    if n.value < x { return search(n.right, x, best_so_far) }
search(root, x, +infinity)

The idea is that if you're exploring the left branch at a node n, you don't need to consider anything to the right of n: n.value is larger than x, and everything to the right is larger than n.value. Similarly, if you're exploring the right branch of a node n, then you can discard everything to the left of n: n.value is smaller than x and everything to the left of n is smaller than n.value.

The code's runtime is bounded by the height of the tree, so is O(log n) if the tree is balanced.

Thanks for your comments on my other answer, Paul. I sat down and actually thought about the problem instead of winging an answer, and I think the following would be a succinct summary of the solution: "The last node at which you recursed down the left subtree is the answer." This means that there would be need to take the min of best so far and the current node's value. Thoughts?

algorithm - Binary Tree Find number nearest and greater than the key -...

algorithm data-structures tree binary-tree
Rectangle 27 7

One interesting example of a binary tree that hasn't been mentioned is that of a recursively evaluated mathematical expression. It's basically useless from a practical standpoint, but it is an interesting way to think of such expressions.

Basically each node of the tree has a value that is either inherent to itself or is evaluated by recursively by operating on the values of its children.

For example, the expression (1+3)*2 can be expressed as:

   / \
  +   2
 / \
1   3

To evaluate the expression, we ask for the value of the parent. This node in turn gets its values from its children, a plus operator and a node that simply contains '2'. The plus operator in turn gets its values from children with values '1' and '3' and adds them, returning 4 to the multiplication node which returns 8.

This use of a binary tree is akin to reverse polish notation in a sense, in that the order in which operations are performed is identical. Also one thing to note is that it doesn't necessarily have to be a binary tree, it's just that most commonly used operators are binary. At its most basic level, the binary tree here is in fact just a very simple purely functional programming language.

What are the applications of binary trees? - Stack Overflow

Rectangle 27 1

To check wheather or not a given Binary Tree is Binary Search Tree here's is an Alternative Approach .

Traverse Tree In Inorder Fashion (i.e. Left Child --> Parent --> Right Child ) , Store Traversed Node Data in a temporary Variable lets say temp , just before storing into temp , Check wheather current Node's data is higher then previous one or not . Then just break it out , Tree is not Binary Search Tree else traverse untill end.

public static boolean isBinarySearchTree(Tree root)
        return false;

        return false;
    return true;

Either subtree can be null. Your algorithm doesn't handle that case correctly.

Difference between binary tree and binary search tree - Stack Overflow

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They can be used as a quick way to sort data. Insert data into a binary search tree at O(log(n)). Then traverse the tree in order to sort them.

What are the applications of binary trees? - Stack Overflow

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The most frequently used implementation of a binary search tree is a red-black tree. The concurrent problems come in when the tree is modified it often needs to rebalance. The rebalance operation can affect large portions of the tree, which would require a mutex lock on many of the tree nodes. Inserting a node into a skip list is far more localized, only nodes directly linked to the affected node need to be locked.

I read Fraser and Harris's latest paper Concurrent programming without locks. Really good stuff if you're interested in lock-free data structures. The paper focuses on Transactional Memory and a theoretical operation multiword-compare-and-swap MCAS. Both of these are simulated in software as no hardware supports them yet. I'm fairly impressed that they were able to build MCAS in software at all.

I didn't find the transactional memory stuff particularly compelling as it requires a garbage collector. Also software transactional memory is plagued with performance issues. However, I'd be very excited if hardware transactional memory ever becomes common. In the end it's still research and won't be of use for production code for another decade or so.

In section 8.2 they compare the performance of several concurrent tree implementations. I'll summarize their findings. It's worth it to download the pdf as it has some very informative graphs on pages 50, 53, and 54.

  • Locking skip lists are insanely fast. They scale incredibly well with the number of concurrent accesses. This is what makes skip lists special, other lock based data structures tend to croak under pressure.
  • transactional skip lists are consistently 2-3 times slower than the locking and non-locking versions.
  • locking red-black trees croak under concurrent access. Their performance degrades linearly with each new concurrent user. Of the two known locking red-black tree implementations, one essentially has a global lock during tree rebalancing. The other uses fancy (and complicated) lock escalation but still doesn't significantly out perform the global lock version.
  • transactional red-black trees are comparable with transactional skip-lists. That was very surprising and very promising. Transactional memory, though slower if far easier to write. It can be as easy as quick search and replace on the non-concurrent version.

Update Here is paper about lock-free trees: Lock-Free Red-Black Trees Using CAS. I haven't looked into it deeply, but on the surface it seems solid.

Not to mention that in a non-degenerate skiplist, about 50% of the nodes should only have a single link which makes insert and delete remarkably efficient.

@Jon, yes and no. There are no known lock-free red-black tree implementations. Fraser and Harris show how a transactional memory based red-black tree is implemented and its performance. Transactional memory is still very much in the research arena, so in production code, a red-black tree will still need to lock large portions of the tree.

I wanted to update this answer. There are currently two lock based efficient binary search trees. One is based on AVL trees ( and the other (Warning! shameless plug) is based on red black trees. See

@deft_code: Intel recently announced an implementation of Transactional Memory via TSX on Haswell. This may prove interesting w.r.t those lock free data structures you mentioned.

algorithm - Skip List vs. Binary Tree - Stack Overflow

algorithm language-agnostic data-structures binary-tree skip-lists
Rectangle 27 3

Binary tree can be anything which has 2 child and 1 parent. It can be implemented as linked list or array, or with your custom API. Once you start to add more specific rules into it, it becomes more specialized tree. Most common known implementation is that, add smaller nodes on left and larger ones on right.

unbalanced and not sorted

For example, in the tree on the left, A has the 6 children {B,C,D,E,F,G}. It can be converted into the binary tree on the right.

Binary Search is technique/algorithm which is used to find specific item on node chain. Binary search works on sorted arrays.

Binary search compares the target value to the middle element of the array; if they are unequal, the half in which the target cannot lie is eliminated and the search continues on the remaining half until it is successful or the remaining half is empty.

A tree representing binary search. The array being searched here is [20, 30, 40, 50, 90, 100], and the target value is 40.

This is one of the implementations of binary tree. This is specialized for searching.

Binary search tree and B-tree data structures are based on binary search.

Binary search trees (BST), sometimes called ordered or sorted binary trees, are a particular type of container: data structures that store "items" (such as numbers, names etc.) in memory.

A binary search tree of size 9 and depth 3, with 8 at the root. The leaves are not drawn.

And finally great schema for performance comparison of well-known data-structures and algorithms applied:

Difference between binary tree and binary search tree - Stack Overflow

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Python doesn't have the quite the extensive range of "built-in" data structures as Java does. However, because Python is dynamic, a general tree is easy to create. For example, a binary tree might be:

class Tree(object):
    def __init__(self):
        self.left = None
        self.right = None = None

You can use it like this:

root = Tree() = "root"
root.left = Tree() = "left"
root.right = Tree() = "right"

The question is tagged with Python3, there's no need to derive class Tree from object then


@platzhirsch: Please read and quote the guideline completely: Google explicitly points out that this is required for Python 2 code to work as expected and recommended to improve compatibility with Py3. Here we're talking about Py3 code. There's no need to do extra, legacy typing.

That's a binary tree, not a general one as asked for.

How can I implement a tree in Python? Are there any built in data stru...

python data-structures tree python-3.x
Rectangle 27 1

What about a binary tree?

public class Node {

    private Node leftChild;
    private Node rightChild;

    public Node(Node leftChild, Node rightChild) {
        this.leftChild = leftChild;
        this.rightChild = rightChild;

    public boolean isLeaf() {
        return this instanceof Leaf;

public class Leaf extends Node {

    private Data data;

    public Leaf(Data data) {
        super(null, null); = data;

         /  \
        /    \
   node1     node2
   /\           /\
  /  \         /  \
leaf1 leaf2  leaf3 leaf4
Leaf leaf1 = new Leaf(new Data());
Leaf leaf2 = new Leaf(new Data());
Leaf leaf3 = new Leaf(new Data());
Leaf leaf4 = new Leaf(new Data());

Node node1 = new Node(leaf1, leaf2);
Node node2 = new Node(leaf3, leaf4);

Node root = new Node(node1, node2);

"data" is a kind of Map, not binary tree.

tree - Multilevel map in Java - Stack Overflow

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Rectangle 27 7

Well, this kind of depends how your Binary Tree is structured. Presumably you have some way of finding the desired leaf node given the root of the tree - simply apply that to both values until the branches you choose diverge.

If you don't have a way to find the desired leaf given the root, then your only solution - both in normal operation and to find the last common node - is a brute-force search of the tree.

algorithm - How to find the lowest common ancestor of two nodes in any...

algorithm binary-tree complexity-theory least-common-ancestor
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A generic tree is a node with zero or more children, each one a proper (tree) node. It isn't the same as a binary tree, they're different data structures, although both shares some terminology.

There isn't any builtin data structure for generic trees in Python, but it's easily implemented with classes.

class Tree(object):
    "Generic tree node."
    def __init__(self, name='root', children=None): = name
        self.children = []
        if children is not None:
            for child in children:
    def __repr__(self):
    def add_child(self, node):
        assert isinstance(node, Tree)
#    *
#   /|\
#  1 2 +
#     / \
#    3   4
t = Tree('*', [Tree('1'),
               Tree('+', [Tree('3'),

How can I implement a tree in Python? Are there any built in data stru...

python data-structures tree python-3.x