Consider the algebraic expression E=(2x+y) (5a-b)^3. Draw the tree T which correspond to the expression E and find the pre-order traversal of T.
Oh, you want me to entertain you with some algebraic humor. Well, you're in luck!
Why did the tree blush when it saw the algebraic expression? Because it couldn't stem the roots of its excitement!
Okay, now let's get down to business. Here's the tree T for the expression E=(2x+y) (5a-b)^3:
E
/ \
(2x+y) (5a-b)^3
|
(5a-b)
|
3
And now, the pre-order traversal of tree T is: E, (2x+y), (5a-b)^3, (5a-b), 3.
Hope that tickled your algebraic funny bone!
To draw the tree T for the expression E=(2x+y)(5a-b)^3, we first need to identify the main operators and their corresponding operands. The main operators are multiplication (*) and exponentiation (^), and the operands are (2x+y) and (5a-b)^3.
Here is the tree T representing the expression E:
```
*
/ \
+ ^
/ \ / \
2x y - 3
/ \
* b
/ \
5 a
```
Now, let's find the pre-order traversal of T. Pre-order traversal means visiting the current node, then recursively visiting its left subtree, and finally recursively visiting its right subtree.
The pre-order traversal of T is as follows:
1. Start from the root: * (multiplication)
2. Visit its left subtree:
2.1. Visit the left child: + (addition)
2.2. Visit its left subtree:
2.2.1. Visit the left child: 2x
2.2.2. Visit the right child: y
2.3. Visit its right subtree:
2.3.1. Visit the left child: - (subtraction)
2.3.2. Visit its left subtree:
2.3.2.1. Visit the left child: * (multiplication)
2.3.2.2. Visit its left subtree:
2.3.2.2.1. Visit the left child: 5
2.3.2.2.2. Visit the right child: a
2.3.3. Visit its right child: b
3. Visit its right subtree:
3.1. Visit the left child: ^
3.2. Visit its left subtree: (5a-b)
3.3. Visit its right child: 3
So, the pre-order traversal of T is:
* + 2x y - * 5 a b ^ (5a-b) 3
To draw the tree T corresponding to the algebraic expression E=(2x+y)(5a-b)^3, we will break down the expression into its components and build the tree from there.
1. Start with the whole expression E as the root of the tree T.
E
2. Break down E into its parts: (2x+y) and (5a-b)^3.
E
/ \
(2x+y) (5a-b)^3
3. Break down (2x+y) further: 2x and y.
E
/ \
(2x+y) (5a-b)^3
/ \
2x y
4. Break down (5a-b)^3: 5a-b and the exponent 3.
E
/ \
(2x+y) (5a-b)^3
/ \
2x y
\
3
5. Break down (5a-b): 5a and b.
E
/ \
(2x+y) (5a-b)^3
/ \
2x y
\
3
/
5a
E
/ \
(2x+y) (5a-b)^3
/ \
2x y
\
3
/ \
5a b
Now we have completed the tree T.
To find the pre-order traversal of T (i.e., visit each node in the order of root, left subtree, right subtree), we start at the root and then recursively traverse the tree.
Pre-order traversal of T: E, (2x+y), 2x, y, (5a-b)^3, 3, 5a, b.
Therefore, the pre-order traversal of T is E, (2x+y), 2x, y, (5a-b)^3, 3, 5a, b.