I was solving a data structure problem and I came across a Binary Tree problem:
Find the maximum node in a binary tree, return the node.
Given the following BST
1
/ \
-5 2
/ \ / \
0 3 -4 -5
I thought that in a Binary Tree, every node's key is greater than all keys in its left sub-tree, and less than all keys in its right sub-tree.
Can a Binary Tree be unsorted ?
A binary search tree is supposed to be sorted and ordered.
But a binary tree can be unsorted like the one you have shown. Binary search tree is a type of binary tree which happened to be sorted.
I thought that in a Binary Tree, every node's key is greater than all
keys in its left sub-tree, and less than all keys in its right
sub-tree.
That's property of Binary Search Tree.
Binary tree can be unsorted, but it isn't suitable for lookup (it doesn't have log(n) complexity).
Can a Binary Tree / Binary Search Tree only have the parent and one of the left or right nodes?
Or is it mandatory to have both the left and right nodes?
Binary implies two. Binary trees have two children pointers, not necessarily with references to anything. Parent references are not necessary - that is up to implementation.
If nodes were mandatory to have both children elements, then the tree would be infinite and leaves would not exist.
So, options for a node are
leaf
left/right child only
both children
Of course, there are alternative ways to think about a "tree". For example, take the binary heap array implementation.
It is not mandatory, you can create other types of links if you want. Define up and down pointers like this.
For a node x define up pointer u(x) as:
if x is the right child, then u(x) is the parent of x
if x is left child or root, then u(x) is the lowest node on the right path of x
For a node x define down pointer d(x) as:
if x's left subtree does not exist, then d(x) = x
if left subtree exists, then d(x) points to the rightmost node of x's left subtree
This is so called ring representation of a tree.
Actually, there is nothing mandatory other than just one rule (i.e. the maximum number of children that any binary tree node can have is two).
Technically:
Even a NULL is considered as a tree node when we define TreeNode *root = NULL.
Though, it has no practical significance, we do pass them as argument while calling insert, delete and display methods.
So, as a coder, assuming that a binary tree will always comprise of at least a parent node can prove fatal if the sanity of root is not checked before processing a binary tree.
A binary search tree is also a binary tree with additional properties. Any Binary tree node can have either 0 or 1 or 2 child nodes. So to answer your question, its okay to have only left or right nodes
I was just coming up with an example for an algorithm I'm working on and realised I don't understand whether the following binary tree is a binary search tree:
3
/ \
2 4
/ \
1 3
The reason I'm unsure is because:
there are duplicates in the tree (3)
Is the rule with binary search trees that all values in the left subtree are smaller than the root?
This is binary search tree..
1) If you traverse it in IN-ORDER you will get the sorted list
2) Every element on the left node is less than root node (in value)
3) As far as the duplication is concerned, it is okay, since the in-order traversal will return 1,2,3,3,4
Yes, the left subtree's nodes' key value is less then the node's key value; and the right subtree's nodes' key value is greater then the node's key value; [here is the binary_serch_tree wiki]: http://en.wikipedia.org/wiki/Binary_search_tree
Given binary tree having L leaf nodes, what will be maximum number of node in that tree? No additional information of the tree type.
If this is a full binary tree, the total number of nodes is equal to (2*L - 1).
For a full tree (meaning each node has either 0 or 2 children) the solution is 2L-1 as said by #mathdan. However if the tree is not full, the answer is actually infinite, easy to prove in an example (L=1):
root
\
node1
\
node2
\
...
\
nodeXYZ
\
leaf
If you decide for a concrete number of nodes N then you can easily include an additional node in the structure, making your claim of N nodes invalid.
Can anyone please explain the difference between binary tree and binary search tree with an example?
Binary tree: Tree where each node has up to two leaves
1
/ \
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.
2
/ \
1 3
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
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 (>).
5
/ \
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".
As everybody above has explained about the difference between binary tree and binary search tree, i am just adding how to test whether the given binary tree is binary search tree.
boolean b = new Sample().isBinarySearchTree(n1, Integer.MIN_VALUE, Integer.MAX_VALUE);
.......
.......
.......
public boolean isBinarySearchTree(TreeNode node, int min, int max)
{
if(node == null)
{
return true;
}
boolean left = isBinarySearchTree(node.getLeft(), min, node.getValue());
boolean right = isBinarySearchTree(node.getRight(), node.getValue(), max);
return left && right && (node.getValue()<max) && (node.getValue()>=min);
}
Hope it will help you. Sorry if i am diverting from the topic as i felt it's worth mentioning this here.
Binary Tree stands for a data structure which is made up of nodes that can only have two children references.
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
A Binary Tree which is not a BST;
5
/ \
/ \
9 2
/ \ / \
15 17 19 21
Binary Search Tree (sorted Tree)
A Binary Search Tree which is also a Binary Tree;
50
/ \
/ \
25 75
/ \ / \
20 30 70 80
Binary Search Tree Node property
Also notify that for any parent node in the BST;
All the left nodes have smaller value than the value of the parent node. In the upper example, the nodes with values { 20, 25, 30 } which are all located on the left (left descendants) of 50, are smaller than 50.
All the right nodes have greater value than the value of the parent node. In the upper example, the nodes with values { 70, 75, 80 } which are all located on the right (right descendants) of 50, are greater than 50.
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.
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.
Binary tree
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.
For example, a labeled binary tree of size 9 and height 3, with a root node whose value is 2. Tree is unbalanced and not sorted.
https://en.wikipedia.org/wiki/Binary_tree
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
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. https://en.wikipedia.org/wiki/Binary_search_algorithm
A tree representing binary search. The array being searched here is [20, 30, 40, 50, 90, 100], and the target value is 40.
Binary search tree
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. https://en.wikipedia.org/wiki/Binary_search_tree
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:
Image taken from Algorithms (4th Edition)
Binary search tree: when inorder traversal is made on binary tree, you get sorted values of inserted items
Binary tree: no sorted order is found in any kind of traversal
A binary tree is a tree whose children are never more than two. A binary search tree follows the invariant that the left child should have a smaller value than the root node's key, while the right child should have a greater value than the root node's key.
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.
Below is an example with Java:
public static boolean isBinarySearchTree(Tree root)
{
if(root==null)
return false;
isBinarySearchTree(root.left);
if(tree.data<temp)
return false;
else
temp=tree.data;
isBinarySearchTree(root.right);
return true;
}
Maintain temp variable outside
A tree can be called as a binary tree if and only if the maximum number of children of any of the nodes is two.
A tree can be called as a binary search tree if and only if the maximum number of children of any of the nodes is two and the left child is always smaller than the right child.
In a Binary search tree, all the nodes are arranged in a specific order - nodes to the left of a root node have a smaller value than its root, and all the nodes to the right of a node have values greater than the value of the root.
In a binary tree, each node has 2 child nodes the left node and the right node.
A Binary Search tree is a special kind of tree in which the nodes are sorted, the left node is smaller than the parent node and the left node is bigger than the parent node.
The binary tree allows duplicate values, Binary search tree doesn't allow duplicate values also carrying out any kind of operation is faster in Binary search tree than in Binary tree since BST is sorted
A binary tree is a tree, in which each node can have at most 2 children.
A binary search tree is a further modification of this, giving a certain relationship to the parent and the two children. Since, there are only two children, i.e., left and right child; the relation is defined as follows:
Left Child <= Parent <= Right Child
It is actually, that simple.