Fix binary search explanation (#138)

* Fix binary search explanation

Elaborate on the possible return values if the element is not found, correct the space complexity claim for recursive implementations.

* Fix typo

Co-authored-by: David Leal <halfpacho@gmail.com>

* add articles

Co-authored-by: David Leal <halfpacho@gmail.com>
Co-authored-by: Vitor Oliveira <vbrazo@gmail.com>

Author: 3 people

en/Search Algorithms/Binary Search.md

@@ -12,17 +12,17 @@ Given a sorted  array of n elements, write a function to search for the index of
 - If it is equal to the target element then return the index
 - Else if it is greater than the target element then consider only the left half of array. (lower index = 0, higher = middle - 1)
 - Else if it is less than the target element then consider only the right half of array. (lower index = middle + 1, higher = length of array)
-- Return -1 if target element is not found in the array (Base Case: If lower index is greater than or equal to higher index)
+- Return -(insertion index + 1) if the target element is not found in the array (If the lower index is greater than or equal to higher index). Some simpler implementations just return -1 if the element is not found. The offset of 1 must be added as the insertion index might be 0 (the searched value might be smaller than all elements in the array). As indexing starts at 0, this must be distinguishable from the case where the target element has the index 0.
 
 #### Time Complexity
 
-O(log n) Worse Case     
+O(log n) Worst Case     
 O(1) Best Case (If middle element of initial array is the target element)
 
 #### Space Complexity
 
 O(1) For iterative approach          
-O(log n) For recursive approach due to recursion call stack
+O(1) For recursive approach *if tail call optimization is used*, O(log n) due to recursion call stack, otherwise
 
 #### Example
 
@@ -35,7 +35,7 @@ array i.e. [1,2,3].
 Here we find the middle element equal to target element so we return its index i.e. 1
 
 target = 9          
-Binary Search should return -1 as 9 is not present in the array
+A simple Binary Search implementation may return -1 as 9 is not present in the array. A more complex one would return the index at which 9 would have to be inserted, which would be `-8` (last position in the array (7) plus one (7+1), negated)`.
 ```
 
 #### Code Implementation Links