# Lab 3: Recursion, Tree Recursion

*Due by 11:59pm on Tuesday, July 7.*

## Starter Files

Download lab03.zip. Inside the archive, you will find starter files for the questions in this lab, along with a copy of the Ok autograder.

## Submission

By the end of this lab, you should have submitted the lab with
`python3 ok --submit`

. You may submit more than once before the
deadline; only the final submission will be graded.
Check that you have successfully submitted your code on
okpy.org.

# Topics

Consult this section if you need a refresher on the material for this lab. It's okay to skip directly to the questions and refer back here should you get stuck.

## Recursion

A recursive function is a function that calls itself in its body, either directly or indirectly. Recursive functions have three important components:

- Base case(s), the simplest possible form of the problem you're trying to solve.
- Recursive case(s), where the function calls itself with a
*simpler argument*as part of the computation. - Using the recursive calls to solve the full problem.

Let's look at the canonical example, `factorial`

.

Factorial, denoted with the

`!`

operator, is defined as:`n! = n * (n-1) * ... * 1`

For example,

`5! = 5 * 4 * 3 * 2 * 1 = 120`

The recursive implementation for factorial is as follows:

```
def factorial(n):
if n == 0:
return 1
return n * factorial(n - 1)
```

We know from its definition that 0! is 1. Since `n == 0`

is the smallest number we
can compute the factorial of, we use it as our base case.
The recursive step also follows from the definition of factorial, i.e., ```
n! =
n * (n-1)!
```

.

The next few questions in lab will have you writing recursive functions. Here are some general tips:

- Paradoxically, to write a recursive function, you must assume that the function
is fully functional before you finish writing it; this is called the
*recursive leap of faith*. - Consider how you can solve the current problem using the solution to
a simpler version of the problem. The amount of work done in a recursive function
can be deceptively little: remember to take the leap of faith and
*trust the recursion*to solve the slightly smaller problem without worrying about how. - Think about what the answer would be in the simplest possible case(s). These will be your base cases - the stopping points for your recursive calls. Make sure to consider the possibility that you're missing base cases (this is a common way recursive solutions fail).
- It may help to write an iterative version first.

## Tree Recursion

A tree recursive function is a recursive function that makes more than one call to itself, resulting in a tree-like series of calls.

A classic example of a tree recursion function is finding the nth Fibonacci number:

```
def fib(n):
if n == 0 or n == 1:
return n
return fib(n - 1) + fib(n - 2)
```

Calling `fib(6)`

results in the following call structure (where `f`

is `fib`

):

Each `f(i)`

node represents a recursive call to `fib`

. Each recursive call
makes another two recursive calls. `f(0)`

and `f(1)`

do not make any recursive
calls because they are the base cases of the function. Because of these base
cases, we are able to terminate the recursion and beginning accumulating the
values.

Generally, tree recursion is effective when you want to explore multiple possibilities or choices at a single step. In these types of problems, you make a recursive call for each choice or for a group of choices. Here are some examples:

- Given a list of paid tasks and a limited amount of time, which tasks should you choose to maximize your pay? This is actually a variation of the Knapsack problem, which focuses on finding some optimal combination of different items.
- Suppose you are lost in a maze and see several different paths. How do you find your way out? This is an example of path finding, and is tree recursive because at every step, you could have multiple directions to choose from that could lead out of the maze.
- Your dryer costs $2 per cycle and accepts all types of coins. How many different combinations of coins can you create to run the dryer? This is similar to the partitions problem from the textbook.

# Required Questions

## What Would Python Display?

### Q1: WWPD: Recursion

Use Ok to test your knowledge with the following "What Would Python Display?" questions:

`python3 ok -q recursion-wwpd -u`

For all WWPD questions, type

`Function`

if you believe the answer is`<function...>`

,`Error`

if it errors,`Infinite`

if there is an infinite loop or infinite recursion, and`Nothing`

if nothing is displayed.

```
>>> def f(a, b):
... if a > b:
... return f(a - 3, 2 * b)
... elif a < b:
... return f(b // 2, a)
... else:
... return b
>>> f(2, 2)
______2
>>> f(7, 4)
______4
>>> f(2, 28)
______8
>>> f(-1, -3)
______Infinite
```

### Q2: WWPD: Journey to the Center of the Earth

Use Ok to test your knowledge with the following "What Would Python Display?" questions:

`python3 ok -q sr-wwpd -u`

For all WWPD questions, type

`Function`

if you believe the answer is`<function...>`

,`Error`

if it errors, and`Nothing`

if nothing is displayed.

```
>>> def crust():
... print("70km")
... def mantle():
... print("2900km")
... def core():
... print("5300km")
... return mantle()
... return core
... return mantle
>>> drill = crust
>>> drill = drill()
______70km
>>> drill = drill()
______2900km
>>> drill = drill()
______5300km
2900km
>>> drill()
______5300km
2900km
Function
```

## Coding Practice

### Q3: Pascal's Triangle

Here's a part of the Pascal's trangle:

```
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
```

Every number in Pascal's triangle is defined as the sum of the item
above it and the item that is directly to the upper left of it. Use
`0`

if the entry is empty.
Define the procedure `pascal(row, column)`

which takes a row and a column,
and finds the value at that position in the triangle. Rows and columns are
zero-indexed; that is, the first row is row 0 instead of 1.

```
def pascal(row, column):
"""Returns a number corresponding to the value at that location
in Pascal's Triangle.
>>> pascal(0, 0)
1
>>> pascal(0, 5) # Empty entry; outside of Pascal's Triangle
0
>>> pascal(3, 2) # Row 4 (1 3 3 1), 3rd entry
3
"""
"*** YOUR CODE HERE ***"
```

Use Ok to test your code:

`python3 ok -q pascal`

### Q4: Repeated, repeated

In Homework 2 you encountered the `repeated`

function, which takes
arguments `f`

and `n`

and returns a function equivalent to the nth
repeated application of `f`

. This time, we want to write `repeated`

recursively. You'll want to use `compose1`

, given below for your
convenience:

```
def compose1(f, g):
""""Return a function h, such that h(x) = f(g(x))."""
def h(x):
return f(g(x))
return h
```

```
def repeated(f, n):
"""Return the function that computes the nth application of func (recursively!).
>>> add_three = repeated(lambda x: x + 1, 3)
>>> add_three(5)
8
>>> square = lambda x: x ** 2
>>> repeated(square, 2)(5) # square(square(5))
625
>>> repeated(square, 4)(5) # square(square(square(square(5))))
152587890625
>>> repeated(square, 0)(5)
5
>>> from construct_check import check
>>> # ban iteration
>>> check(HW_SOURCE_FILE, 'repeated',
... ['For', 'While'])
True
"""
"*** YOUR CODE HERE ***"
```

Use Ok to test your code:

`python3 ok -q repeated`

### Q5: Num eights

Write a recursive function `num_eights`

that takes a positive integer `x`

and
returns the number of times the digit 8 appears in `x`

.
*Use recursion - the tests will fail if you use any assignment statements.*

```
def num_eights(x):
"""Returns the number of times 8 appears as a digit of x.
>>> num_eights(3)
0
>>> num_eights(8)
1
>>> num_eights(88888888)
8
>>> num_eights(2638)
1
>>> num_eights(86380)
2
>>> num_eights(12345)
0
>>> from construct_check import check
>>> # ban all assignment statements
>>> check(HW_SOURCE_FILE, 'num_eights',
... ['Assign', 'AugAssign'])
True
"""
"*** YOUR CODE HERE ***"
```

Watch the hints video below for somewhere to start:

Use Ok to test your code:

`python3 ok -q num_eights`

### Q6: Ping-pong

The ping-pong sequence counts up starting from 1 and is always either counting
up or counting down. At element `k`

, the direction switches if `k`

is a
multiple of 8 or contains the digit 8. The first 30 elements of the ping-pong
sequence are listed below, with direction swaps marked using brackets at the
8th, 16th, 18th, 24th, and 28th elements:

Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | [8] | 9 | 10 | 11 | 12 | 13 | 14 | 15 | [16] | 17 | [18] | 19 | 20 | 21 | 22 | 23 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PingPong Value | 1 | 2 | 3 | 4 | 5 | 6 | 7 | [8] | 7 | 6 | 5 | 4 | 3 | 2 | 1 | [0] | 1 | [2] | 1 | 0 | -1 | -2 | -3 |

Index (cont.) | [24] | 25 | 26 | 27 | [28] | 29 | 30 |
---|---|---|---|---|---|---|---|

PingPong Value | [-4] | -3 | -2 | -1 | [0] | -1 | -2 |

Implement a function `pingpong`

that returns the nth element of the ping-pong
sequence *without using any assignment statements*.

You may use the function `num_eights`

, which you defined in the previous question.

*Use recursion - the tests will fail if you use any assignment statements.*

Hint: If you're stuck, first try implementing`pingpong`

using assignment statements and a`while`

statement. Then, to convert this into a recursive solution, write a helper function that has a parameter for each variable that changes values in the body of the while loop.

```
def pingpong(n):
"""Return the nth element of the ping-pong sequence.
>>> pingpong(8)
8
>>> pingpong(10)
6
>>> pingpong(15)
1
>>> pingpong(21)
-1
>>> pingpong(22)
-2
>>> pingpong(30)
-2
>>> pingpong(68)
0
>>> pingpong(69)
-1
>>> pingpong(80)
0
>>> pingpong(81)
1
>>> pingpong(82)
0
>>> pingpong(100)
-6
>>> from construct_check import check
>>> # ban assignment statements
>>> check(HW_SOURCE_FILE, 'pingpong', ['Assign', 'AugAssign'])
True
"""
"*** YOUR CODE HERE ***"
```

Watch the hints video below for somewhere to start:

Use Ok to test your code:

`python3 ok -q pingpong`