# Recursion

Print Cheatsheet

### Base Case of a Recursive Function

A recursive function should have a base case with a condition that stops the function from recursing indefinitely. In the example, the base case is a condition evaluating a negative or zero value to be true.

``````function countdown(value)
if value is negative or zero
print "done"
otherwise if value is greater than zero
print value
call countdown with (value-1)``````

### Recursive Step in Recursive Function

A recursive function should have a recursive step which calls the recursive function with some input that brings it closer to its base case. In the example, the recursive step is the call to `countdown()` with a decremented value.

``````def countdown(value):
if value <= 0:
print("done")
else:
print(value)
countdown(value-1)  #recursive step ``````

### What is Recursion

Recursion is a strategy for solving problems by defining the problem in terms of itself. A recursive function consists of two basic parts: the base case and the recursive step.

### Call Stack in Recursive Function

Programming languages use a facility called a call stack to manage the invocation of recursive functions. Like a stack, a call stack for a recursive function calls the last function in its stack when the base case is met.

### Big-O Runtime for Recursive Functions

The big-O runtime for a recursive function is equivalent to the number of recursive function calls. This value varies depending on the complexity of the algorithm of the recursive function. For example, a recursive function of input N that is called N times will have a runtime of O(N). On the other hand, a recursive function of input N that calls itself twice per function may have a runtime of O(2^N).

### Weak Base Case in Recursive Function

A recursive function with a weak base case will not have a condition that will stop the function from recursing, causing the function to run indefinitely. When this happens, the call stack will overflow and the program will generate a stack overflow error.

### Execution Context of a Recursive Function

An execution context of a recursive function is the set of arguments to the recursive function call. Programming languages use execution contexts to manage recursive functions.

### Stack Overflow Error in Recursive Function

A recursive function that is called with an input that requires too many iterations will cause the call stack to get too large, resulting in a stack overflow error. In these cases, it is more appropriate to use an iterative solution. A recursive solution is only suited for a problem that does not exceed a certain number of recursive calls.

For example, `myfunction()` below throws a stack overflow error when an input of 1000 is used.

``````def myfunction(n):
if n == 0:
return n
else:
return myfunction(n-1)

myfunction(1000)  #results in stack overflow error``````

### Fibonacci Sequence

A Fibonacci sequence is a mathematical series of numbers such that each number is the sum of the two preceding numbers, starting from 0 and 1.

``Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...``

### Binary Search Tree

In Python, a binary search tree is a recursive data structure that makes sorted lists easier to search. Binary search trees:

• Reference two children at most per tree node.
• The “left” child of the tree must contain a value lesser than its parent.
• The “right” child of the tree must contain a value greater than it’s parent.
``````     5
/ \
/   \
3     8
/ \   / \
2   4 7   9``````

### Call Stack Construction in While Loop

A call stack with execution contexts can be constructed using a `while` loop, a `list` to represent the call stack and a `dictionary` to represent the execution contexts. This is useful to mimic the role of a call stack inside a recursive function.

### Recursion and Nested Lists

A nested list can be traversed and flattened using a recursive function. The base case evaluates an element in the list. If it is not another list, the single element is appended to a flat list. The recursive step calls the recursive function with the nested list element as input.

``````def flatten(mylist):
flatlist = []
for element in mylist:
if type(element) == list:
flatlist += flatten(element)
else:
flatlist += element
return flatlist

print(flatten(['a', ['b', ['c', ['d']], 'e'], 'f']))
# returns ['a', 'b', 'c', 'd', 'e', 'f']``````

### Fibonacci Recursion

Computing the value of a Fibonacci number can be implemented using recursion. Given an input of index N, the recursive function has two base cases – when the index is zero or 1. The recursive function returns the sum of the index minus 1 and the index minus 2.

The Big-O runtime of the Fibonacci function is O(2^N).

``````def fibonacci(n):
if n <= 1:
return n
else:
return fibonacci(n-1) + fibonacci(n-2)``````

### Modeling Recursion as Call Stack

One can model recursion as a `call stack` with `execution contexts` using a `while` loop and a Python `list`. When the `base case` is reached, print out the call stack `list` in a LIFO (last in first out) manner until the call stack is empty.

Using another `while` loop, iterate through the call stack `list`. Pop the last item off the list and add it to a variable to store the accumulative result.

Print the result.

``````def countdown(value):
call_stack = []
while value > 0 :
call_stack.append({"input":value})
print("Call Stack:",call_stack)
value -= 1
print("Base Case Reached")
while len(call_stack) != 0:
print("Popping {} from call stack".format(call_stack.pop()))
print("Call Stack:",call_stack)
countdown(4)
'''
Call Stack: [{'input': 4}]
Call Stack: [{'input': 4}, {'input': 3}]
Call Stack: [{'input': 4}, {'input': 3}, {'input': 2}]
Call Stack: [{'input': 4}, {'input': 3}, {'input': 2}, {'input': 1}]
Base Case Reached
Popping {'input': 1} from call stack
Call Stack: [{'input': 4}, {'input': 3}, {'input': 2}]
Popping {'input': 2} from call stack
Call Stack: [{'input': 4}, {'input': 3}]
Popping {'input': 3} from call stack
Call Stack: [{'input': 4}]
Popping {'input': 4} from call stack
Call Stack: []
'''``````

### Recursion in Python

In Python, a recursive function accepts an argument and includes a condition to check whether it matches the base case. A recursive function has:

• Base Case - a condition that evaluates the current input to stop the recursion from continuing.
• Recursive Step - one or more calls to the recursive function to bring the input closer to the base case.
``````def countdown(value):
if value <= 0:   #base case
print("done")
else:
print(value)
countdown(value-1)  #recursive case ``````

### Build a Binary Search Tree

To build a binary search tree as a recursive algorithm do the following:

``````BASE CASE:
If the list is empty, return "No Child" to show that there is no node.

RECURSIVE STEP:
1. Find the middle index of the list.
2. Create a tree node with the value of the middle index.
3. Assign the tree node's left child to a recursive call with the left half of list as input.
4. Assign the tree node's right child to a recursive call with the right half of list as input.
5. Return the tree node.
``````
``````def build_bst(my_list):
if len(my_list) == 0:
return "No Child"

middle_index = len(my_list) // 2
middle_value = my_list[middle_index]

print("Middle index: {0}".format(middle_index))
print("Middle value: {0}".format(middle_value))

tree_node = {"data": middle_value}
tree_node["left_child"] = build_bst(my_list[ : middle_index])
tree_node["right_child"] = build_bst(my_list[middle_index + 1 : ])

return tree_node

sorted_list = [12, 13, 14, 15, 16]
binary_search_tree = build_bst(sorted_list)
print(binary_search_tree)``````