## CS 314 Assignment 6: Basic Trees

### Due: March 13, 2015.

Files: Cons.java   test6.lsp

All of the following functions that operate on trees should be recursive. Some functions that you may need are provided in the file Cons.java . This assignment may be done in Java or in Lisp.

1. Write a function maxbt(Object tree) that finds the maximum value in a binary tree of Integer. For this function, we will assume that both the first and rest of a Cons can be subtrees. We will assume that every element of the tree is a Cons (which can be tested with consp and then cast as (Cons)), null, or Integer. We will assume that every Integer is greater than Integer.MIN_VALUE.
Example: (maxbt '((1 7) (-3 2) (((8)))) ) = 8

2. An algebraic expression can be written as a linked list tree: an expression is either a number (Integer or Double), a symbol (String), or a list (op lhs rhs) where op is an operator (String) and lhs (left-hand side) and rhs (right-hand side) are expressions. The functions op, lhs and rhs are provided. Write a function Cons vars(Object expr) that returns a set of the variables in an expression. Note that the op and numbers are not variables. The result is a set (list) of variables in any order; the union function is provided.
Example: (vars '(= f (* m a))) = (f m a)

3. Write a function boolean occurs(Object value, Object tree) that tests whether the value occurs anywhere in the expression tree. We will assume that the value has a .equals() method and is not a Cons.
Example: (occurs 'm '(= f (* m a))) = true

4. Write a function Integer eval(Object tree) that evaluates (finds the value of) an expression tree where the leaves are Integer. The value of an Integer expression is the expression itself. If the expression is a Cons, first find the value of its arguments (lhs and rhs), then perform the operation denoted by the op. The possible operations are + - * / expt . (expt x n) raises x to the power n; a function pow is provided. Note that - could have either one operand (minus) or two operands (difference); all other operations are assumed to be binary.
Note: if you use Lisp, call your function myeval; eval is the Lisp interpreter.
Example: (eval '(+ 3 (* 5 7))) = 38

5. Write a function Integer eval(Object tree, Cons bindings) that evaluates an expression tree where the leaves are Integer or are variables whose values are given in the bindings. bindings is an association list, ((var value) ...), that gives values for variables. This version of eval is an easy extension of the previous one. The function assoc is provided.
Example: (eval '(+ 3 (* 5 b)) '((b 7))) = 38

6. Write a function Cons english(Object tree) that translates an expression tree into a list of English words (Strings). If the tree is a leaf, the translation is just a list of that tree. For an expression, make a list containing "the", an appropriate English word for the operator, and the operands connected by "of" and "and". The english program should only use operations such as cons, list, append, first, rest, op, lhs, rhs etc.; it should not use any String operations.
Example: (english '(+ 3 (* b 7))) = (the sum of 3 and the product of b and 7)

7. There is a close relationship between programming languages and trees. A compiler performs parsing, which converts a character string in a programming language to a tree. Unparsing, converting a tree to a program, is also useful. Write a function String tojava(Object tree) that translates an expression tree to a String that is a line of Java code. The line of Java code should be terminated by a semicolon character. We will assume that the expression can contain the operators = + - * / as well as single-argument functions such as sin.

Operators have precedence, which determines the order in which operations are performed when an expression is not parenthesized. We will assume that = has precedence 1, + - have precedence 5, and * / have precedence 6. A subexpression needs to be parenthesized if its precedence is less than or equal to the precedence of its surroundings; otherwise, it should not be parenthesized. Make an auxiliary function that includes precedence as an argument. The starting precedence can be 0, so that any operator will be higher in precedence.
Example: (tojava '(= x (* (+ a b) c))) = "x=(a+b)*c;"

We will assume that a unary minus should always be parenthesized, and that it has a precedence of 6.

For functions that are not in the operator list, such as sin, make the name be Math. followed by the function name, and make a function call form. For example, (sin x) would become "Math.sin(x)" .