Question

C++

What are the differences and similarities between Binary Search Tree and Binary Searches of arrays.

Answer 1

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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.

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:

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 )
    }
  }
}

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