A recursive function is a function that is defined in terms of itself. Consider this recursive
def factorial(n): if n == 0 or n == 1: return 1 else: return n * factorial(n - 1)
Although we haven’t finished defining
factorial, we are still able to call it since the function body is not evaluated until
the function is called. When
n is 0 or 1, we just return 1. This is known as the base case, and it prevents the
function from infinitely recursing. Now we can compute
factorial(2) in terms of
factorial(3) in terms of
factorial(4) – well, you get the idea.
There are three common steps in a recursive definition:
- Figure out your base case: The base case is usually the simplest input possible to the function. For example,
factorial(0)is 1 by definition. You can also think of a base case as a stopping condition for the recursion. If you can’t figure this out right away, move on to the recursive case and try to figure out the point at which we can’t reduce the problem any further.
- Make a recursive call with a simpler argument: Simplify your problem, and assume that a recursive call for this new
problem will simply work. This is called the “leap of faith”. For
factorial, we reduce the problem by calling
factorial(n - 1).
- Use your recursive call to solve the full problem: Remember that we are assuming the recursive call works. With the result of the recursive call, how can you solve the original problem you were asked? For
factorial, we just multiply (n − 1)! by n.
Another way to understand recursion is by separating out two things: "internal correctness" and not running forever (known as "halting").
A recursive function is internally correct if it is always does the right thing assuming that every recursive call does the right thing.
Consider this alternative recursive
def factorial(n): # WRONG! if n == 2: return n return n * factorial(n-1)
It is internally correct, since 2! = 2 and n! = n ∗ (n − 1)! are both true statements.
factorial does not halt on all inputs, since
factorial(1) results in a call to
factorial(0), and then to
factorial(-1) and so on.
A recursive function is correct if and only if it is both internally correct and halts; but you can check each property separately. The "recursive leap of faith" is temporarily placing yourself in a mindset where you only check internal correctness.
Q1: Recursion Environment Diagram
Draw an environment diagram for the following code:
def rec(x, y): if y > 0: return x * rec(x, y - 1) return 1 rec(3, 2)
Imagine you were writing the documentation for this function. Come up with a line that describes what the function does:
You may also want to watch this video walkthrough.
Note: This problem is meant to help you understand what really goes on when we make the "recursive leap of faith". However, when approaching or debugging recursive functions, you should avoid visualizing them in this way for large or complicated inputs, since the large number of frames can bes quite unwieldy and confusing. Instead, think in terms of the three step process - base case, recursive call, solving the full problem.
Q2: Merge Numbers
Write a procedure
merge(n1, n2) which takes numbers with digits in decreasing order
and returns a single number with all of the digits of the two, in decreasing order.
Any number merged with 0 will be that number (treat 0 as having no digits). Use recursion.
Your Answer Run in 61A Code
Hint: If you can figure out which number has the smallest digit out of both, then we know that the resulting number will have that smallest digit, followed by the merge of the two numbers with the smallest digit removed.
def merge(n1, n2): """ Merges two numbers by digit in decreasing order >>> merge(31, 42) 4321 >>> merge(21, 0) 21 >>> merge (21, 31) 3211 """if n1 == 0: return n2 elif n2 == 0: return n1 elif n1 % 10 < n2 % 10: return merge(n1 // 10, n2) * 10 + n1 % 10 else: return merge(n1, n2 // 10) * 10 + n2 % 10
Q3: Is Prime
Write a function
is_prime that takes a single argument
n and returns
n is a prime number and
False otherwise. Assume
n > 1. We implemented
this in Discussion 1 iteratively, now time to do it recursively!
Your Answer Run in 61A Code
Hint: You will need a helper function! Remember helper functions are useful if you need to keep track of more variables than the given parameters, or if you need to change the value of the input.
def is_prime(n): """Returns True if n is a prime number and False otherwise. >>> is_prime(2) True >>> is_prime(16) False >>> is_prime(521) True """def helper(i): if i > (n ** 0.5): # Could replace with i == n return True elif n % i == 0: return False return helper(i + 1) return helper(2)
Q4: (Tutorial) Warm Up: Recursive MultiplicationThese exercises are meant to help refresh your memory of topics covered in lecture and/or lab this week before tackling more challenging problems.
Write a function that takes two numbers
n and returns their product. Assume
n are positive integers. Use recursion, not
5 * 3 = 5 + (5 * 2) = 5 + 5 + (5 * 1).
For the base case, what is the simplest possible input for multiply?
For the recursive case, what does calling
multiply(m - 1, n) do? What does calling
multiply(m, n - 1) do? Do we prefer one over the other?
nless than the total, while the second will calculate a value that is
mless. Either recursive call will work, but only
multiply(m, n - 1)is used in this solution.
Your Answer Run in 61A Code
Challenge: Try to implement the multiply function tail recursively.
def multiply(m, n): """ Takes two positive integers and returns their product using recursion. >>> multiply(5, 3) 15 """if n == 1: return m else: return m + multiply(m, n - 1) # TAIL RECURSIVE SOLUTION def multiply_helper(x, y, result): if y == 0: return result else: return multiply_helper(x, y - 1, result + x) return multiply_helper(m, n, 0)
Q5: (Tutorial) Recursive Hailstone
hailstone function from Homework 1.
First, pick a positive integer
n as the start. If
n is even, divide it by 2.
n is odd, multiply it by 3 and add 1. Repeat this process until
n is 1.
Write a recursive version of
hailstone that prints out the values of the sequence and returns the number of steps.
Your Answer Run in 61A Code
Hint: When taking the recursive leap of faith, consider both the return value and side effect of this function.
def hailstone(n): """Print out the hailstone sequence starting at n, and return the number of elements in the sequence. >>> a = hailstone(10) 10 5 16 8 4 2 1 >>> a 7 """print(n) if n == 1: return 1 elif n % 2 == 0: return 1 + hailstone(n // 2) else: return 1 + hailstone(3 * n + 1)
Consider a function that requires more than one recursive call. A simple example is the recursive
def fib(n): if n == 0: return 0 elif n == 1: return 1 else: return fib(n - 1) + fib(n - 2)
This type of recursion is called
tree recursion, because it makes more than one recursive call in its recursive case. If we draw out the recursive calls, we see the recursive calls in the shape of an upside-down tree:
We could, in theory, use loops to write the same procedure. However, problems that are naturally solved using tree recursive procedures are generally difficult to write iteratively. It is sometimes the case that a tree recursive problem also involves iteration: for example, you might use a while loop to add together multiple recursive calls.
As a general rule of thumb, whenever you need to try multiple possibilities at the same time, you should consider using tree recursion.
Q6: Count Stair Ways
Imagine that you want to go up a flight of stairs that has
n steps, where
n is a positive integer. You can either take 1 or 2 steps each time. In this question, you'll write a function
count_stair_ways that solves this problem. Before you code your approach, consider these questions.
How many different ways can you go up this flight of stairs?
What’s the base case for this question? What is the simplest input?
fibonaccineeds 2 base cases -- to cover both recursive calls. ALTERNATE Solution: Our first base case is where there are no steps left. This means that we took an action in the previous recursive step that led to our goal of reaching the top. Our second base case is where we have overstepped. This means that the action we took is not valid, as it caused us to step over our goal.
count_stair_ways(n - 1) and
count_stair_ways(n - 2) represent?
count_stair_ways(n - 1)represents the number of different ways to go up the last
n−1stairs (this is the case where we take 1 step as our move).
count_stair_ways(n - 2)represents the number of different ways to go up the last
n−2stairs (this is the case where we take 2 steps as our move). Our base cases will take care of the situations where there are no steps left or we overstepped.
Fill in the code for
def count_stair_ways(n): """Returns the number of ways to climb up a flight of n stairs, moving either 1 step or 2 steps at a time. >>> count_stair_ways(4) 5 """if n == 1: return 1 elif n == 2: return 2 return count_stair_ways(n-1) + count_stair_ways(n-2)
Check out these video walkthroughs explaining the approach:
Q7: (Tutorial) Count K
Consider a special version of the
count_stair_ways problem, where instead of taking 1 or 2 steps, we are able to take up to and including
k steps at a time. Write a function
count_k that figures out the number of paths for this scenario. Assume
k are positive.
def count_k(n, k): """ Counts the number of paths up a flight of n stairs when taking up to and including k steps at a time. >>> count_k(3, 3) # 3, 2 + 1, 1 + 2, 1 + 1 + 1 4 >>> count_k(4, 4) 8 >>> count_k(10, 3) 274 >>> count_k(300, 1) # Only one step at a time 1 """if n == 0: return 1 elif n < 0: return 0 else: total = 0 i = 1 while i <= k: total += count_k(n - i, k) i += 1 return total
Here's a tree of the calls made to