a) Represent the expression ((x+2) ↑3) * (y-(3+x))-5 using a binary tree. Write this expression

A) Represent the expression ((x+2) ↑3) * (y-(3+x))-5 using a...

Key Concepts

Binary Expression Trees

A binary expression tree is a tree data structure used to represent arithmetic expressions. In this representation, each internal node denotes an operator and each leaf node denotes an operand. This structure allows one to clearly visualize the hierarchical organization of operators and operands, facilitating operations such as expression evaluation, conversion between notations, and simplifying expressions.

Prefix Notation

Prefix notation, also known as Polish notation, is a form of writing expressions where the operator precedes all of its operands. This format eliminates the need for parentheses to denote order of operations, as it inherently encodes the order in which operations are to be performed. It is especially useful for computer processing in recursive algorithms and stack-based evaluations.

Postfix Notation

Postfix notation, or Reverse Polish Notation, is an arithmetic expression format where the operator follows all of its operands. Similar to prefix notation, it removes the need for parentheses as the order of operations is unambiguously determined by the position of operators. This notation is advantageous in computer science for its straightforward evaluation using stacks.

Infix Notation

Infix notation is the most familiar form of writing expressions in which the operator is placed between the operands. Although it closely mimics standard mathematical notation, it typically requires the use of parentheses or adherence to operator precedence rules to accurately represent the intended order of operations. This makes it slightly more complex to process algorithmically compared to prefix or postfix forms.

Transcript

00:01 In part a, we're given an expression, and we're asked to represent this expression using a binary tree.

00:11 So we're given the expression x plus 2 the third power times y minus 3 plus x minus 5.

00:22 So to construct the binary tree to represent this expression, first we construct some subtrees.

00:33 So we'll have the sub -tree representing x plus 2.

00:40 We'll also have the sub -tree representing 3 plus x.

00:54 Next, we'll create the sub -tree representing x plus 2 to the third power, and we'll also construct the sub -tree representing y -minus 3 -plus -x.

01:43 Using these two sub -trees, first we'll construct the sub -tree x plus 2 to the third -power.

01:51 Times y minus 3 plus x and finally using this subtree will construct the binary trade for the full expression we have a rooted binary tree with root minus and on the left we're going to have our sub tree with root star and star has children power and minus exponentiation and subtraction extenation has children plus and three, plus has children x and two, minus has children y and plus, addition, i should say.

04:02 Addition has children three and x.

04:09 And then we see that minus also has a child five.

04:13 So this is the binary tree representing this expression.

04:27 Now in part b, we're asked to write the expression using prefix notation.

04:31 So let's make a prefix proversal of this tree that we've constructed so i'm going to start off with our root minus followed by the subtree with root star star as we determined as children exponentiation say multiplication as children exponentiation and subtraction expansiation is children addition addition and three.

05:34 In addition, it's children, x and two.

05:37 There are a fast ways to do this if you'd like them just being explicit here.

05:42 And subtraction has children y and addition.

05:47 Addition has children three and x.

05:55 This is followed by the subtree or the prefix, the prefix traversal of the subtree with root 5, which is simply 5.

06:10 So in the next step, we have minus, followed by the subtree, or the prefix traversal of the subtree with root exponentiation.

06:31 So we have exponentiation.

06:36 I guess we should really have the root multiplication first, then exponentiation.

06:44 Exponitiation has children addition and three, addition has children x and two, followed by the prefix traversal of the subtree with root subtraction.

07:09 You see that subtraction has children y and addition...

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