Discussion 4: Tree Recursion

disc04.pdf

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Getting Started

Say your name and your favorite tree (a particular tree or a kind of tree) in honor of today's topic: tree recursion.

Definition: Tree recursive functions are functions that call themselves more than once.

In this discussion, don't use a Python interpreter to run code until you are confident your solution is correct. Figure things out and check your work by thinking about what your code will do. Not sure? Talk to your group!

[New] Recursion takes practice. Please don't get discouraged if you're struggling to write recursive functions. Instead, every time you do solve one (even with help or in a group), make note of what you had to realize to make progress. Students improve through practice and reflection.

[For Fun] This emoticon of a guy in a cowboy hat is valid Python: o[:-D]

>>> o = [2, 0, 2, 4]
>>> [ o[:-D] for D in range(1,4) ]
[[2, 0, 2], [2, 0], [2]]

🤠

Tree Recursion

For the following questions, don't start trying to write code right away. Instead, start by describing the recursive case in words. Some examples:

In fib from lecture, the recursive case is to add together the previous two Fibonacci numbers.
In double_eights from lab, the recursive case is to check for double eights in the rest of the number.
In count_partitions from lecture, the recursive case is to partition n-m using parts up to size m and to partition n using parts up to size m-1.

Q1: Insect Combinatorics

An insect is inside an m by n grid. The insect starts at the bottom-left corner (1, 1) and wants to end up at the top-right corner (m, n). The insect can only move up or to the right. Write a function paths that takes the height and width of a grid and returns the number of paths the insect can take from the start to the end. (There is a closed-form solution to this problem, but try to answer it with recursion.)

Insect grids.

In the 2 by 2 grid, the insect has two paths from the start to the end. In the 3 by 3 grid, the insect has six paths (only three are shown above).

Hint: What happens if the insect hits the upper or rightmost edge of the grid?

Presentation Time: Once your group has converged on a solution, it's time to practice your ability to describe why your recursive case is correct. Nominate someone and have them present to the group for practice. Then, tell this description to your TA for feedback (on Zoom if your TA is remote).

Tree Recursion with Lists

[New] Some of you already know list operations that we haven't covered yet, such as append. Don't use those today. All you need are list literals (e.g., [1, 2, 3]), item selection (e.g., s[0]), list addition (e.g., [1] + [2, 3]), len (e.g., len(s)), and slicing (e.g., s[1:]). Use those! There will be plenty of time for other list operations when we introduce them next week.

The most important thing to remember about lists is that a non-empty list s can be split into its first element s[0] and the rest of the list s[1:].

>>> s = [2, 3, 6, 4]
>>> s[0]
2
>>> s[1:]
[3, 6, 4]

Q2: Max Product

Implement max_product, which takes a list of numbers and returns the maximum product that can be formed by multiplying together non-consecutive elements of the list. Assume that all numbers in the input list are greater than or equal to 1.

First try multiplying the first element by the max_product of everything after the first two elements (skipping the second element because it is consecutive with the first), then try skipping the first element and finding the max_product of the rest. To find which of these options is better, use max.

A great way to get help is to talk to the course staff!

Complete this sentence together: "The recursive case is to choose the larger of ... and ..."

Description Time: Now try to complete this sentence together: "The recursive case is to choose the larger of _ and _." When you're done, see how your answer compares to ours.

The recursive case is to choose the larger of the largest product that includes the first but not the second element and the largest product that does not include the first element.

Q3: Sum Fun

Implement sums(n, m), which takes a total n and maximum m. It returns a list of all lists:

that sum to n,
that contain only positive numbers up to m, and
in which no two adjacent numbers are the same.

Two lists with the same numbers in a different order should both be returned.

Here's a recursive approach that matches the template below: build up the result list by building all lists that sum to n and start with k, for each k from 1 to m. For example, the result of sums(5, 3) is made up of three lists:

[[1, 3, 1]] starts with 1,
[[2, 1, 2], [2, 3]] start with 2, and
[[3, 2]] starts with 3.

Hint: Use [k] + s for a number k and list s to build a list that starts with k and then has all the elements of s.

>>> k = 2
>>> s = [4, 3, 1]
>>> [k] + s
[2, 4, 3, 1]

k is the first number in a list that sums to n, and rest is the rest of that list, so build a list that sums to n.

Call sums to build all of the lists that sum to n-k so that they can be used to construct lists that sum to n by putting a k on the front.

Here is where you ensure that "no two adjacent numbers are the same." Since k will be the first number in the list you're building, it must not be equal to the first element of rest (which will be the second number in the list you're building).

If you get stuck (which many groups do), ask for help!

Document the Occasion

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