[chore] add markdown version of recursion notebooks

Author: gitgik

recursion/bst_diameter.md

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+### Problem
+Given the root of a Binary Search Tree, find its diameter.
+
+Sample input:
+
+```
+         10
+       /   \ 
+      5     15
+     / \     \
+    2   5    22
+   /
+   1
+```
+
+Expected output:
+
+```
+6 => (We get 6 from counting 1, 2, 5, 10, 15, 22)
+```
+
+### Solution
+The diameter of a tree T is the largest (max) of the following quantities:
+
+- the diameter of T’s left subtree.
+- the diameter of T’s right subtree.
+- the longest path between leaves that goes through the root of T (this can be computed from the heights of the subtrees of T)
+
+
+```python
+class Node:
+    def __init__(self, value, left=None, right=None):
+        self.value = value
+        self.left = left
+        self.right = right
+        
+def height(root):
+    if root is None:
+        return 0
+    return 1 + max(height(root.left), height(root.right))
+
+def diameter(root):
+    # base case: tree is empty
+    if root is None:
+        return 0
+    
+    lheight = height(root.left)
+    rheight = height(root.right)
+    
+    ldiameter = diameter(root.left)
+    rdiameter = diameter(root.right)
+    
+    return max(lheight + rheight + 1, max(ldiameter, rdiameter))
+```
+
+
+```python
+bst = Node(6, Node(3, Node(1), Node(4, Node(2))), Node(8, Node(7), Node(9)))
+```
+
+
+```python
+diameter(bst)
+```
+
+
+
+
+    6
+
+

recursion/count_sets_adding_to_target.md

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+## Problem
+Given an array of non-negative integers, find the number of all the sets that add up to a given target.
+
+Sample input:
+```
+[1, 2, 4, 5, 7],  Target = 7
+```
+
+The output is `3`, i.e {1, 2, 4}, {2, 5} and {7} all add up to 7
+
+
+```python
+def count_sets(arr, total):
+    return helper(arr, total, len(arr) - 1, {})
+
+def helper(arr, total, i, mem) -> int:
+    key = f"{total}:{i}"
+    if key in mem:
+        return mem[key]
+    
+    if total == 0:
+        return 1
+    elif total < 0:
+        return 0
+    elif i < 0:
+        return 0 
+    elif total < arr[i]:
+        to_return = helper(arr, total, i - 1, mem)
+    else:
+        to_return = helper(arr, total - arr[i], i - 1, mem) + helper(arr, total, i - 1, mem)
+    mem[key] = to_return
+    return to_return
+```
+
+
+```python
+count_sets([1, 2, 4, 5, 7], 7)
+```
+
+
+
+
+    3
+
+

recursion/fibonacci.md

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+```python
+def fibonacci(n, memoize={1:0, 2: 1}):
+    """
+    Nth fibonacci number using memoization.
+    
+    O(n) space and time
+    """
+    if n < 0:
+        raise TypeError('Value of n must be a positive integer')
+    if n in memoize:
+        return memoize[n]
+    else:
+        memoize[n] = fibonacci(n - 1, memoize) + fibonacci(n - 2, memoize)
+        return memoize[n]
+
+
+def efficient_fibonacci(n):
+    """
+    nth fibonacci number iteratively.
+
+    Complexity:
+        O(n) time,
+        O(1) space, since we are only storing two array values at any given time.
+    """
+    last_two = [0, 1]
+
+    counter = 3
+    while counter <= n:
+        next_fibonacci = sum(last_two)
+        last_two = [last_two[1], next_fibonacci]
+        counter += 1
+    # the else clause caters for the edge case of the first fibo number == 0
+    return last_two[1] if n > 1 else last_two[0]
+```
+
+
+```python
+efficient_fibonacci(12)
+```
+
+
+
+
+    89
+
+
+
+
+```python
+
+```

recursion/greatest_common_divisor.md

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+## Problem
+Given n numbers, find the greatest common denominator between them.
+
+For example, given the numbers [42, 56, 14], return 14
+
+## Solution
+Because the greatest common divisor is associative, gcd of multiple numbers: say `a, b, c` is equivalent to 
+```
+gcd(gcd(ab),c)
+```
+By definition, if the divisor divides gcd(a, b) and c, it must divide a and b as well.
+
+The GCD of multiple numbers `a, b, ... n` can be obtained by **iteratively computing the GCD of `a and b`, and GCD of the result of that with `c` and so on.**
+
+
+```python
+def gcd(numbers):
+    n = numbers[0]
+    for num in numbers[1:]:
+        n = _gcd(n, num)
+    return n
+```
+
+A naive implementation might try every integer from 1 to min(a, b) and see of it divides the larger:
+
+
+```python
+def _gcd_naive(a, b):
+    # find smallest value, largest value of the two
+    smaller, larger = min(a, b), max(a,b)
+    # iteratively check every integer from the smallest value to 1.
+    for divisor in range(smaller, 0, -1):
+        if larger % divisor == 0:
+            return divisor
+```
+
+A more efficient method is the Euclidean algorithm which follows a recursive formula:
+
+```
+gcd(a, 0) = a
+gcd(a, b) = gcd(b, a % b)
+```
+
+
+```python
+def _gcd(a, b):
+    if b == 0:
+        return a
+    return _gcd(b, a % b)
+```
+
+
+```python
+# time to test it out
+gcd([42, 56, 14, 7])
+```
+
+
+
+
+    7
+
+
+
+Here's a more memory-efficient method that works bottom up:
+
+
+```python
+def _gcd(a, b):
+    while b:
+        a, b = b, a % b
+    return a
+
+# test it out
+gcd([0, 64, 256, 128])
+```
+
+
+
+
+    64
+
+
+
+For more details check out this video tutorial on how [Euclidean algorithm](https://www.youtube.com/watch?v=p5gn2hj51hs) works.
+
+

recursion/matrix_maximum_coins.md

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+### Problem 
+You are given a 2-d matrix where each cell represents number of coins in that cell. Assuming we start at matrix[0][0], and can only move right or down, find the maximum number of coins you can collect by the bottom right corner.
+
+For example, in this matrix
+
+```python
+0 3 1 1
+2 0 0 4
+1 5 3 1
+```
+
+The most we can collect is 0 + 2 + 1 + 5 + 3 + 1 = 12 coins.
+
+### Approach
+Consider the following matrix,
+
+```
+0 2 3
+5 4 9
+6 7 1
+```
+
+- If we are at 1, the most coins we can collect is ` 1 `
+- If we start at 7, the most coins we'll co is `7 + 1`
+- If we start at 9, the most coins we can have is `9 + 1`
+- If we start at 4, the most coins we can collect is `max(7 + 1,  9 + 1)`
+
+This is a recursive problem. 
+
+The base case will be when we get to the bottom-right corner of the matrix – the most amount of coins we can collect at that point is that cell.
+This leads to the following solution:
+
+
+```python
+def collect_coins(matrix, r=0, c=0, cache=None):
+    if cache is None:
+        cache = {}
+    
+    is_bottom = (r == len(matrix) - 1)
+    is_rightmost = (c == len(matrix[0]) - 1)
+    
+    if (r, c) not in cache:
+        if is_bottom and is_rightmost:
+            cache[r, c] = matrix[r][c]
+        elif is_bottom:
+            cache[r, c] = matrix[r][c] + collect_coins(matrix, r, c + 1, cache)
+        elif is_rightmost:
+            cache[r, c] = matrix[r][c] + collect_coins(matrix, r + 1, c, cache)
+        else:
+            cache[r, c] = matrix[r][c] + max(collect_coins(matrix, r, c + 1, cache), collect_coins(matrix, r + 1, c, cache))
+    return cache[r, c]
+```
+
+
+```python
+matrix = [
+  [0, 3, 1, 1],
+  [2, 0, 0, 4],
+  [1, 5, 3, 1]
+]
+collect_coins(matrix)
+```
+
+
+
+
+    12
+
+

recursion/min_max.md

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+### Problem
+Given an array of numbers of length N, find both the minimum and maximum using less than 2 * (N - 2) comparisons.
+
+
+```python
+def min_max(array):
+    if len(array) == 1:
+        return array[0], array[0]
+    elif len(array) == 2:
+        return (array[0], array[1]) if array[0] < array[1] else (array[1], array[0])
+    else:
+        mid = len(array) // 2
+        left_min, left_max = min_max(array[:mid])
+        right_min, right_max = min_max(array[mid:])
+        return min(left_min, right_min), max(left_max, right_max)
+
+```
+
+
+```python
+min_max([1, 2, 3, 4])
+```
+
+
+
+
+    (1, 4)
+
+
+
+For array of size N, we are breaking down the problem into two subproblems of size `N / 2` plus `2` comparisons. 
+
+```python
+# Recursively break down array
+[4, 2, 7, 5, -1, 3, 6] 
+[[4, 2, 7, 5], [-1, 3, 6]]
+[[[4, 2], [7, 5]], [[-1, 3], [6]]]
+
+# Base case: reorder so that smaller comes before larger
+[[[2, 4], [5, 7]], [[-1, 3], [6, 6]]]
+
+# Merge to find min and max
+[[min(2, 5), max(4, 7)], [min(-1, 6), max(3, 6)]]
+[min(2, -1), max(7, 6)]
+[-1, 7] 
+```
+
+
+```python
+
+```