Difference between binary search and binary search tree?

For those who came here to quickly check which one to use. In addition to the answers,posted above, I would like to add complexities with respect to the operations for both of these techniques.

Binary search Tree:

Search: θ(log(n)), Worst case (O(n)) for unbalanced BST,

Insert of node: θ(log(n)) , Worst case (O(n)) for unbalanced BST

Deletion of node: θ(log(n)), , Worst case (O(n)) for unbalanced BST

Balanced Binary search Tree:

Search: log(n),

Insert of node: O(log(n))

Deletion of node: O(log(n))

Binary Search on sorted array:

Search: O(log(n)) But,

Insertion of node: Not possible if array is statically allocated and already full. Otherwise O(n) ( O(n) for moving larger items of array to their adjacent right)

Deletion of node: O(log(n)) + O(n). (So it would be O(log(n)) for finding position of deletion + O(n) for moving larger items of array to their adjacent left)

So based on your requirements, you can choose whether you need quick inserts and deletes. If you don't need these, then keeping things in sorted array will work for you, as array will take less memory compared to tree

No, they're not the same.

Binary search tree:

• A tree data structure
• Each node has up to 2 children
• The left subtree of a node contains only nodes with keys less than the node's key
• The right subtree of a node contains only nodes with keys greater than the node's key

Binary search:

• An algorithm that works on a sorted data structure (usually, but not necessarily, an array) and, at each step, looking at the value in the middle and recursing to either the left or the right, depending on whether the target value is smaller or greater than the value in the middle (or stopping if it's equal).

And, of course, a data structure is:

A particular way of storing and organizing data in a computer so that it can be used efficiently.

While an algorithm is:

A step-by-step procedure for calculations.

The search process in a binary search tree (where we look for a specific value in the tree) can be thought of as similar to (or an instance of, depending on your definitions and whether you're using a balanced BST) binary search, since this also looks at the 'middle' element and recurses either left or right, depending on the result of the comparison between that value and the target value.

Binary Search Trees

A node in a binary tree is a data structure that has an element, and a reference to two other binary trees, typically called the left and right subtrees. I.e., a node presents an interface like this:

Node:
  element  (an element of some type)
  left     (a binary tree, or NULL)
  right    (a binary tree, or NULL)

A binary search tree is a binary tree (i.e., a node, typically called the root) with the property that the left and right subtrees are also binary search trees, and that all the elements of all the nodes in the left subtree are less than the root's element, and all the elements of all the nodes in the right subtree are greater than the root's element. E.g.,

     5
    / \
   /   \
  2     8
 / \   / \
1   3 6   9

Binary Search

Binary search is an algorithm for finding an element in binary search tree. (It's often expressed as a way of searching an ordered collection, and this is an equivalent description. I'll describe the equivalence afterward.) It's this:

search( element, tree ) {
  if ( tree == NULL ) {
    return NOT_FOUND
  }
  else if ( element == tree.element ) {
    return FOUND_IT
  }
  else if ( element < tree.element ) {
    return search( element, tree.left )
  }
  else {
    return search( element, tree.right )
  }
}

This is typically an efficient method of search because at each step, you can remove half the search space. Of course, if you have a poorly balanced binary search tree, it can be inefficient (it can degrade to linear search). For instance, it has poor performance in a tree like:

3
 \
  4
   \
    5
     \
      6

Binary Search on Arrays

Binary search is often presented as a search method for sorted arrays. This does not contradict the description above. In fact, it highlights the fact that we don't actually care how a binary search tree is implemented; we just care that we can take an object and do three things with it: get a element, get a left sub-object, and get a right sub-object (subject, of course, to the constraints about the elements in the left being less than the element, and the elements in the right being greater, etc.).

We can do all three things with a sorted array. With a sorted array, the "element" is the middle element of the array, the left sub-object is the subarray to the left of it, and the right sub-object is the subarray to the right of it. E.g., the array

[1 3 4 5 7 8 11]

corresponds to the tree:

     5
    / \
   /   \
  3     8
 / \   / \
1  4  7   11

Thus, we can write a binary search method for arrays like this:

search( element, array, begin, end ) {
  if ( end <= begin ) {
    return NOT_FOUND
  }
  else { 
    midpoint = begin+(end-begin)/2
    a_element = array[midpoint]
    if ( element == midpoint ) {
      return FOUND_IT
    }
    else if ( element < midpoint ) {
      return search( element, array, begin, midpoint )
    }
    else {
      return search( element, array, midpoint, end )
    }
  }
}

Conclusion

As often presented, binary search refers to the array based algorithm presented here, and binary search tree refers to a tree based data structure with certain properties. However, the properties that binary search requires and the properties that binary search trees have make these two sides of the same coin. Being a binary search tree often implies a particular implementation, but really it's a matter of providing certain operations and satisfying certain constraints. Binary search is an algorithm that functions on data structures that have these operations and meet these constraints.