Computer Science Fundamentals 107. A binary tree applica.on - Expression Trees

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1 Computer Science Fundamentals 107 A binary tree applica.on - Expression Trees

2 Parse trees 2 Originally parsing came from the study of language breaking up sentences into components. We can represent the structure of a sentence using a parse tree. Compilers and interpreters are two examples of applica.ons which use parsing.

3 Expression trees An expression tree can be used to store and evaluate an algebraic expression. The leaves are the values and the non-leaf nodes are the operators. To evaluate the expression we work from the boeom up:

4 Tokens 4 Parsers recognise tokens. A token is a symbol made up of one or more characters (commonly tokens are separated by white spaces). For an expression tree the tokens are numbers, operators and parentheses. For example, for the following expression: * 5 the tokens are, +, 6, -, 2, * and 5.

5 Expression Tree example 5 The expression tree on the right represents the following expression: * 5 # 6 %* 2 5 We will be dealing with a fully parenthesised expression, e.g., ( + (6 (2 * 5)))

6 Expression trees parsing rules As we iden.fy tokens in the expression we can apply rules to what we should do in order to create the tree. 6 When dealing with a fully parenthesised expression, + a ler parenthesis "(" starts a subtree a right parenthesis ")" finishes a subtree # 6 %* ( + (6 (2 * 5))) 2 5

7 7 Crea=ng an expression tree 4 rules 1. If the current token is a '(', add a new node as the ler child of the current node, and descend to the ler child. 2. If the current token is in the list ['+',' ','*','/'], set the root value of the current node to the operator represented by the current token. Add a new node as the right child of the current node and descend to the right child.. If the current token is a number, set the root value of the current node to the number and return to the parent. 4. If the current token is a ')', go to the parent of the current node.

8 8 Crea=ng an expression tree 4 rules 1. If the current token is a '(', add a new node as the ler child of the current node, and descend to the ler child. ( + ( 6 ( 2 * 5 ) ) )!!!

9 9 Crea=ng an expression tree 4 rules. If the current token is a number, set the root value of the current node to the number and return to the parent. ( + ( 6 ( 2 * 5 ) ) )!! "

10 10 Crea=ng an expression tree 4 rules 2. If the current token is in the list ['+',' ','*','/'], set the root value of the current node to the operator represented by the current token. Add a new node as the right child of the current node and descend to the right child. ( + ( 6 ( 2 * 5 ) ) ) " "

11 11 Crea=ng an expression tree 4 rules 1. If the current token is a '(', add a new node as the ler child of the current node, and descend to the ler child. ( + ( 6 ( 2 * 5 ) ) ) " " "

12 12 Crea=ng an expression tree 4 rules. If the current token is a number, set the root value of the current node to the number and return to the parent. ( + ( 6 ( 2 * 5 ) ) ) " " " 6

13 1 Crea=ng an expression tree 4 rules 2. If the current token is in the list ['+',' ','*','/'], set the root value of the current node to the operator represented by the current token. Add a new node as the right child of the current node and descend to the right child. ( + ( 6 ( 2 * 5 ) ) ) " 6 6 "

14 14 Crea=ng an expression tree 4 rules 1. If the current token is a '(', add a new node as the ler child of the current node, and descend to the ler child. ( + ( 6 ( 2 * 5 ) ) ) 6 " 6 " "

15 15 Crea=ng an expression tree 4 rules. If the current token is a number, set the root value of the current node to the number and return to the parent ( + ( 6 ( 2 * 5 ) ) ) 6 " " 6 " "2

16 16 Crea=ng an expression tree 4 rules 2. If the current token is in the list ['+',' ','*','/'], set the root value of the current node to the operator represented by the current token. Add a new node as the right child of the current node and descend to the right child. ( + ( 6 ( 2 * 5 ) ) ) 6 " "2 6 "* "2 "

17 17 Crea=ng an expression tree 4 rules. If the current token is a number, set the root value of the current node to the number and return to the parent. ( + ( 6 ( 2 * 5 ) ) ) 6 "* "2 " 6 "* "2 "5

18 18 Crea=ng an expression tree 4 rules 4. If the current token is a ')', go to the parent of the current node. ( + ( 6 ( 2 * 5 ) ) ) 6 "* "2 "5 6 "* "2 "5

19 19 Crea=ng an expression tree 4 rules 4. If the current token is a ')', go to the parent of the current node. ( + ( 6 ( 2 * 5 ) ) ) 6 "* "2 "5 6 "* "2 "5

20 20 Crea=ng an expression tree 4 rules 4. If the current token is a ')', go to the parent of the current node. ( + ( 6 ( 2 * 5 ) ) ) expression_tree 6 "* "2 "5 6 "* "2 "5

21 Crea=ng an expression tree code 21 Before looking at the first token in the expression, create an empty stack, create a binary tree with one empty node, and push the reference to this expression tree onto the stack. Ini.alise the current node to point to the expression tree. parent_stack = Stack() expression_tree = RefBinaryTree('') parent_stack.push(expression_tree) = expression_tree parent_stack expression_tree! At the end of the algorithm when the last ')' is reached, the only element in the stack is a reference to the whole expression tree. ( + ( 6 ( 2 * 5 ) ) ) expression_tree parent_stack 6 "* "2 "5

22 Crea=ng an expression tree code 22 When crea.ng the expression tree, the algorithm requires us to to move up the tree to the parent node, so we will use a stack to keep track of all the parents. from RefBinaryTree import RefBinaryTree from Stack import Stack def build_expression_tree(parenthesized_expression): list_of_tokens = parenthesized_expression.split() parent_stack = Stack() expression_tree = RefBinaryTree('') parent_stack.push(expression_tree) = expression_tree. def main(): Ini.alise tree to an empty node Push reference to the tree onto the stack. This will be the last item removed from the stack at the end of the evalua.on. expression_tree = build_expression_tree("( ( ) * )") print(expression_tree) main()

23 Crea=ng an expression tree code 2 1. If the current token is a '(', add a new node as the ler child of the current node, and descend to the ler child. def build_expression_tree(parenthesized_expression): list_of_tokens = parenthesized_expression.split() parent_stack = Stack() expression_tree = RefBinaryTree('') parent_stack.push(expression_tree) current_tree = expression_tree for token in list_of_tokens: if token == '(':.insert_ler('') parent_stack.push() =.get_ler_subtree() Create an empty node ler child, push the reference to the parent node onto the stack, and move to the ler child.

24 Crea=ng an expression tree code If the current token is in the list ['+',' ','*','/'], set the root value of the current node to the operator represented by the current token. Add a new node as the right child of the current node and descend to the right child.. def build_expression_tree(parenthesized_expression): expression_tree = RefBinaryTree('') parent_stack.push(expression_tree) = expression_tree for token in list_of_tokens: elif token in ['+', '-', '*', '/']:.set_value(token).insert_right('') parent_stack.push() =.get_right_subtree() Put the operator into the node, add a new right child, push the reference to the parent node onto the stack, and move to the right child.

25 Check if token is a number The following is a useful func.on for checking if the token (a string) can be converted into a float. 25 def is_number(token): try: float(token) except: return False return True def main(): print(is_number("")) print(is_number(")) main() True False

26 Crea=ng an expression tree code 26. If the current token is a number, set the root value of the current node to the number and return to the parent. 4. If the current token is a ')', go to the parent of the current node. def build_expression_tree(parenthesized_expression): expression_tree = RefBinaryTree('') parent_stack.push(expression_tree) = expression_tree for token in list_of_tokens: elif is_number(token):.set_value(float(token)) = parent_stack.pop() elif token == ')': = parent_stack.pop() Put the number into the node, pop the parent stack, and move to the parent node. Pop the parent stack, and move to the parent node.

27 Crea=ng an expression tree code 27 The build_expression_tree() func.on, can be used to create an expression tree from a fully parenthesised algebraic expression. from RefBinaryTree import RefBinaryTree from Stack import Stack def build_expression_tree(parenthesized_expression): expression_tree = RefBinaryTree('') for token in list_of_tokens: if token == '(':. elif token == :. return expression_tree def main(): expression_tree = build_expression_tree("( ( ) * )") print(expression_tree) main()

28 Why expression trees? 28 They are easy to evaluate! + +!1 # #4 6 "* 6 %10 "2 "5 = -1

29 Evaluate the expression tree 29 To evaluate the subtree under a node: if the node has children, e.g., (in this case the node holds an operator)!2!5 return the result of applying the operator on the ler and right subtrees!* else the node holds a number, e.g., return the number.!5 6 Note that the examples only have binary operators.

30 Evaluate the expression tree 0 If the tree is a single node, return the value. Otherwise, apply the node opera.on (+-*/) on the result of evalua.ng the two subtrees and return the result. expression_tree!+ 6!*!2!5

31 Evaluate the expression tree 1 expression_tree (+ is not a single node) so evaluate ler child, evaluate right child and return + - eventually returns 16!+ becomes:!+ 6!* 6!10!2!5 Evaluate ler child - returns 6 6 Evaluate right child!* (* is not a single node) so evaluate ler child,!2!5 evaluate right child and return * - eventually returns 10 Evaluate ler child - returns 2 Evaluate right child - returns 5!2!5

32 Perform the required opera=on The following returns the result of applying ", "-", "*' or "/" to the two operands. 2 def perform_opera=on(operator, num_ler, num_right): if operator == ": return num_ler + num_right if operator == "-": return num_ler - num_right if operator == "*": return num_ler * num_right return num_ler / num_right def main(): print(perform_opera=on(", 2, 5)) print(perform_opera=on("/", 12, 2)) main() 7 6.0

33 Evaluate the expression tree code Use recursion to evaluate the expression tree. def evaluate_tree(expression_tree): token = expression_tree.get_value() ler = expression_tree.get_ler_subtree() right = expression_tree.get_right_subtree() if ler and right: ler_value = evaluate_tree(ler) right_value = evaluate_tree(right) return perform_opera.on(token, ler_value, right_value) else: return token def main(): expression_tree = build_expression_tree("( ( ) * )") print(evaluate_tree(expression_tree)) main()

Binary Tree Applications. Chapter 6.6

Binary Tree Applications. Chapter 6.6 Binary Tree Applications Chapter 6.6 1 Parse Trees What is parsing? Originally from language study The breaking up of sentences into component parts e.g. noun phrase In computing compilers and interpreters