Sample Questions: SICP Section 2.2

Hierarchical Data and the Closure Property

## Cons structures

1. How many cons cells make up the structure with the print value of `(1 (2 3) 4)`?
1. 4
2. 6
3. 5
4. 3
1. How many cons cells make up the structure with the print value of `(1 (2 3) . 4)`?
1. 5
2. 4
3. 6
4. 3
1. How many cons cells make up the structure with the print value of `(1 (2 . 3) . 4)`?
1. 4
2. 6
3. 3
4. 5
1. How many cons cells make up the structure with the print value of `(1 (2 . 3) 4)`?
1. 5
2. 4
3. 6
4. 3
1. What is the minimum number of cons cells needed to bundle up 4 numbers into a single cons structure?
1. 6
2. 5
3. 4
4. 3
1. What is the minimum number of cons cells needed to bundle up 4 numbers into a single list structure (i.e. no nested lists)?
1. 6
2. 5
3. 4
4. 3

## Heirarchical structures

1. Draw a cons structure in the style of the text that represents a binary search tree with the elements 3, 1, 2, 5 inserted in that order. Use nils to indicate missing children.
1. Draw a cons structure in the style of the text that represents a trinary search tree with the elements 3, 1, 2, 1, 5, 1 inserted in that order. A trinary search tree behaves just as a binary search tree except duplicates are inserted as middle children. Use nils to indicate missing children.
1. Consider the singly-linked list $2\to 5\to 3\to$nil. Draw a cons cell representation of this list, assuming there is only a head pointer. Use the style of the text. Use the variable items to point to this list.
1. Consider the singly-linked list $2\to 5\to 3\to$nil. Draw a cons cell representation of this list, assuming there is a head and a tail pointer. Use the style of the text. Use the variable items to point to this list.

## Conventional interfaces

1. (3 points) Define a function named flatten that takes deeply nested list and returns a flat list of all the elements in the original list. For example, `(flatten '((a (b)) c))` evaluates to `(a b c)`. You may find the functions pair? or atom? useful.
1. ount the number of.Sum the odd values in a deeply nested list. Pick from the components flatten, map, keep, remove, accumulate, append, and odd?. Assume the name of the incoming list is items.
1. Suppose I wish to find the product of all the prime numbers from 1 to n? Pick from the components enumerate, map, keep, remove, accumulate, and prime?. Start with n.
1. Suppose I wish to collect all the non-prime numbers from 1 to n? Pick from the components enumerate, map, keep, accumulate, and prime?. Start with n. Note: remove is not available.
1. Sum of all the even numbered squares of the numbers in a given list named items. Pick from the components enumerate, map, remove, accumulate, square, and even?. Note: keep is not available.
1. Collect all the odd numbered squares of the numbers in a given list named items. Pick from the components enumerate, map, remove, accumulate, square, and even?.
1. Collect the squares of all the Mersenne primes generated from a list of exponents, named raises. A Mersenne prime has the form ${2}^{n}-1$. Pick from the components enumerate, map, keep, remove, accumulate, prime?, square, and expt.

## Nested mappings

1. (2 points) Collect all the pairs $\left(i,j\right)$ such that $0\le i and $0\le j. Pick from the components enumerate, map, keep, remove, accumulate, and expand. The expand function takes a list, a single item, and a location, and creates a list of lists composed each of the list items and the single item. For example, `(expand '(1 4 2) 0 'back)` evaluates to `((1 0) (4 0) (2 0))`, while `(expand '(1 4 2) 0 'front)` evaluates to `((0 1) (0 4) (0 2))`. Start with n.
1. (2 points) Collect all the pairs $\left(i,j\right)$ such that $0\le i and $0\le j. Pick from the components enumerate, map, keep, remove, accumulate, and expand. The expand function takes a list, a single item, and a location, and creates a list of lists composed each of the list items and the single item. For example, `(expand '(1 4 2) 0 'back)` evaluates to `((1 0) (4 0) (2 0))`, while `(expand '(1 4 2) 0 'front)` evaluates to `((0 1) (0 4) (0 2))`. Start with n.
1. (2 points) Collect all the pairs $\left(i,j\right)$ such that $0\le i and $i\le j. Pick from the components enumerate, map, keep, remove, accumulate, and expand. The expand function takes a list, a single item, and a location, and creates a list of lists composed each of the list items and the single item. For example, `(expand '(1 4 2) 0 'back)` evaluates to `((1 0) (4 0) (2 0))`, while `(expand '(1 4 2) 0 'front)` evaluates to `((0 1) (0 4) (0 2))`. Start with n.

1. (3 points) Define a function named index that retrieves the ${i}^{th}$ element of a given list. Do not perform any error checking.
1. (3 points) Define a function named ntail that returns the ${n}^{th}$ tail of a given list. When passed a value of 0, ntail should return the given list; when passed a value of 1, ntail should return the cdr of the list, and so on. Do not perform any error checking.