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1. Write a single Boolean LISP function, called TREE-CONTAINS, which takes two arguments N and
TREE, and checks whether number N appears in the ordered tree TREE.
For example,
(TREE-CONTAINS 3 ‘((1 2 3) 7 8)) returns T
(TREE-CONTAINS 4 ‘((1 2 3) 7 8)) returns NIL
2. Write a single LISP function, called TREE-MIN, which takes one argument TREE, and returns the
minimum number appearing in the ordered tree TREE.
For example,
(TREE-MIN ‘((1 2 3) 7 8)) returns 1
3. Write a single LISP function, called TREE-ORDER, which takes one argument TREE, and returns an
pre-ordered list of the numbers appearing in the ordered tree TREE.
For example,
(TREE-ORDER 3) returns (3)
(TREE-ORDER ‘((1 2 3) 7 8)) returns (7 2 1 3 8)
4. Write a single LISP function, called SUB-LIST, that takes a list L and two non-negative integers START
and LEN, and returns the sub-list of L starting at position START and having length LEN. Assume that the
first element of L has position 0.
For example,
(SUB-LIST ‘(a b c d) 0 3) returns (a b c)
(SUB-LIST ‘(a b c d) 3 1) returns (d)
(SUB-LIST ‘(a b c d) 2 0) return s NIL
5. Write a single LISP function, called SPLIT-LIST, that takes a list L, and returns a list of two lists L1 and
L2, in that order, such that
– L is the result of appending L1 and L2;
– Length of L1 minus length of L2 is 0 or 1.
For example,
(SPLIT-LIST ‘(a b c d)) returns ((a b) (c d))
(SPLIT-LIST ‘(a b c d e)) returns ((a b c) (d e)) NOTE: ((a b) (c d e)) is incorrect;
(SPLIT-LIST ‘(a b c d e f)) returns ((a b c) (d e f))
You can call the function SUB-LIST from SPLIT-LIST.
A binary tree is one in which each node has 0 or 2 children. A node that has 0 child is called a leaf node. A
node that has 2 children is called an internal node. A binary tree can be represented as follows:
– A leaf node N is represented by atom N;
– An internal node N is represented by a list (L R), where L represents the left child of N and R
represents the right child of N.
6. Write a single LISP function, called BTREE-HEIGHT, which takes a binary tree TREE, and returns the
height of TREE. Note that the height of a binary tree is defined as the length of the longest path from the
root node to the farthest leaf node.
For example,
(BTREE-HEIGHT 1) returns 0
(BTREE-HEIGHT ‘(1 2)) returns 1
(BTREE-HEIGHT ‘(1 (2 3))) returns 2
(BTREE-HEIGHT ‘((1 2) (3 4))) returns 2
(BTREE-HEIGHT ‘((1 (2 3)) ((4 5) (6 7)))) returns 3
(BTREE-HEIGHT ‘(((1 2) (3 4)) ((5 6) (7 8)))) returns 3
Here is the binary tree for ‘((1 (2 3)) ((4 5) (6 7)))
7. Write a single LISP function, called LIST2BTREE, that takes a non-empty list of atoms LEAVES, and
returns a binary tree such that
– The tree leaves are the elements of LEAVES;
– For any internal (non-leaf) node in the tree, the number of leaves in its left branch minus the
number of leaves in its right branch is 0 or 1.
For example,
(LIST2BTREE ‘(1)) returns 1
(LIST2BTREE ‘(1 2)) returns (1 2)
(LIST2BTREE ‘(1 2 3)) returns ((1 2) 3)
(LIST2BTREE ‘(1 2 3 4)) returns ((1 2) (3 4))
(LIST2BTREE ‘(1 2 3 4 5 6 7)) returns (((1 2) (3 4)) ((5 6) 7))
(LIST2BTREE ‘(1 2 3 4 5 6 7 8)) returns (((1 2) (3 4)) ((5 6) (7 8)))
You can call the function SPLIT-LIST from LIST2BTREE.
8. Write a single LISP function, called BTREE2LIST, that takes a binary tree TREE as input, and returns a
list of atoms (assume TREE follows the constraints we defined earlier).
– As the input is a binary tree, each node has at most 2 children;
– This function is the inverse of LIST2BTREE. That is, (BTREE2LIST (LIST2BTREE X)) = X for all
lists of atoms X.
For example,
(BTREE2LIST 1) returns (1)
(BTREE2LIST ‘(1 2)) returns (1 2)
(BTREE2LIST ‘((1 2) 3)) returns (1 2 3)
(BTREE2LIST ‘((1 2) (3 4))) returns (1 2 3 4)
(BTREE2LIST ‘(((1 2) (3 4)) ((5 6) 7))) returns (1 2 3 4 5 6 7)
(BTREE2LIST ‘(((1 2) (3 4)) ((5 6) (7 8)))) returns (1 2 3 4 5 6 7 8)
9. Write a single Boolean LISP function, called IS-SAME, that takes two LISP expressions E1 and E2
whose atoms are all numbers, and checks whether the expressions are identical. In this question, you can
only use ‘=‘ to test equality (you cannot use ‘equal’). Recall that a LISP expression is either an atom or a
list of LISP expressions.
For example,
(IS-SAME ‘((1 2 3) 7 8) ‘((1 2 3) 7 8)) returns T
(IS-SAME ‘(1 2 3 7 8) ‘((1 2 3) 7 8)) returns NIL