1. Review: recursion Tree traversals
October 05, 2017
Cinda Heeren / Geoffrey Tien

2. Cinda Heeren / Geoffrey Tien
Rabbits!
...and recursion
What happens when you put a pair of rabbits in a field?
More rabbits!
Let’s model the rabbit population, with a few assumptions:
Newly-born rabbits take one month to reach maturity and mate
Each pair of rabbits produces another pair of rabbits one month after mating
Rabbits never die
October 05, 2017
Cinda Heeren / Geoffrey Tien

3. Cinda Heeren / Geoffrey Tien
More rabbits...
How many rabbit pairs are there after 5 months?
Month 1 2 3 4 5 6
Month 1: start – 1 pair
Month 2: first pair are now mature and mate – 1 pair 1 1 2 3 5 8
Month 3: first pair give birth to a pair of babies – original pair + baby pair = 2 pairs
Month 5: the 3 pairs from month 4, and two new pairs – 5 pairs
Month 4: first pair give birth to another pair of babies, pair born in month 3 are now mature – 3 pairs
Month 6: the 5 pairs from month 5, and three new pairs – 8 pairs
And so on...
October 05, 2017
Cinda Heeren / Geoffrey Tien

4. Cinda Heeren / Geoffrey Tien
Fibonacci series
The nth number in the Fibonacci series, fib(n), is:
0 if n = 0, and 1 if n = 1
fib(n – 1) + fib(n – 2) for any n > 1
e.g. what is fib(23)
Easy if we only knew fib(22) and fib(21)
The answer is fib(22) + fib(21)
What happens if we actually write a function to calculate Fibonacci numbers like this?
October 05, 2017
Cinda Heeren / Geoffrey Tien

5. Calculating the Fibonacci series
Let’s write a function just like the formula
fib(n) = 0 if n = 0, 1 if n = 1,
otherwise fib(n) = fib(n – 1) + fib(n – 2)
int fib(int n) {
if (n <= 1)
return max(0, n);
else
return fib(n-1) + fib(n-2); }
The function calls itself
October 05, 2017
Cinda Heeren / Geoffrey Tien

6. Cinda Heeren / Geoffrey Tien
Recursive functions
The Fibonacci function is recursive
A recursive function calls itself
Each call to a recursive method results in a separate call to the method, with its own input
Recursive functions are just like other functions
The invocation (e.g. parameters, etc.) is pushed onto the call stack
And removed from the call stack when the end of a method or a return statement is reached
Execution returns to the previous method call
October 05, 2017
Cinda Heeren / Geoffrey Tien

7. Recursive function anatomy
Recursive functions do not use loops to repeat instructions
But use recursive calls, in if statements
Recursive functions consist of two or more cases, there must be at least one
Base case, and
Recursive case
October 05, 2017
Cinda Heeren / Geoffrey Tien

8. Cinda Heeren / Geoffrey Tien
Recursion cases
The base case is a smaller problem with a known solution
This problem’s solution must not be recursive
Otherwise the function may never terminate
There can be more than one base case
And base cases may be implicit
The recursive case is the same problem with smaller input
The recursive case must include a recursive function call
There can be more than one recursive case
October 05, 2017
Cinda Heeren / Geoffrey Tien

9. Cinda Heeren / Geoffrey Tien
Analysis of fib(5)
int fib(int n) {
if (n <= 1)
return max(0, n);
else
return fib(n-1) + fib(n-2); } 5
fib(5) 2 3
fib(4)
fib(3) 1 2 1 1
fib(3)
fib(2)
fib(2)
fib(1) 1 1 1 1
fib(2)
fib(1)
fib(1)
fib(0)
fib(1)
fib(0) 1
Later in the course we will explain how this is an extremely inefficient way to compute the Fibonacci series
fib(1)
fib(0)
October 05, 2017
Cinda Heeren / Geoffrey Tien

10. Example Base cases Recursive case Recursive linear search
Target is found, or the end of the array is reached
Recursive case
Target not found, search subarray starting from next element
// Recursive linear search
int recLinSearch(int arr[], int next, int sz, int x) {
if (next >= sz) // end of array reached
return -1;
else if (x == arr[next]) // target found
return next;
else // not found, search from different starting index
return recLinSearch(arr, next + 1, sz, x); }
October 05, 2017
Cinda Heeren / Geoffrey Tien

11. Back on Topic – binary tree traversal
A traversal algorithm for a binary tree visits each node in the tree
Typically, it will do something while visiting each node!
Traversal algorithms are naturally recursive
There are three traversal methods
inOrder
preOrder
postOrder
October 05, 2017
Cinda Heeren / Geoffrey Tien

12. inOrder traversal algorithm
void inOrder(Node* nd) {
if (nd != NULL)
inOrder(nd->leftchild);
visit(nd);
inOrder(nd->rightchild); }
The visit function would do whatever the purpose of the traversal is (e.g. print the data value of the node).
October 05, 2017
Cinda Heeren / Geoffrey Tien

13. Cinda Heeren / Geoffrey Tien
preOrder Traversal
visit(nd);
preOrder(nd->leftchild); 1
preOrder(nd->rightchild); 17 2 6
visit
visit 13 27
preOrder(left)
preOrder(left)
preOrder(right)
preOrder(right) 3 5 7 8
visit
visit
visit 9 16 20 39
preOrder(left)
preOrder(left)
preOrder(left)
preOrder(right)
preOrder(right)
preOrder(right)
visit 4
preOrder(left)
preOrder(right) 11
visit
preOrder(left)
preOrder(right)
October 05, 2017
Cinda Heeren / Geoffrey Tien

14. Cinda Heeren / Geoffrey Tien
postOrder traversal
postOrder(nd->leftchild);
postOrder(nd->rightchild); 8
visit(nd); 17 4 7
postOrder(left)
postOrder(left) 13 27
postOrder(right)
postOrder(right)
visit
visit 2 3 5 6
postOrder(left)
postOrder(left)
postOrder(left) 9 16 20 39
postOrder(right)
postOrder(right)
postOrder(right)
visit
visit
visit
postOrder(left) 1
postOrder(right)
visit
postOrder(left) 11
postOrder(right)
visit
October 05, 2017
Cinda Heeren / Geoffrey Tien

15. Cinda Heeren / Geoffrey Tien
Exercise
What will be printed by an in-order traversal of the tree?
preOrder? postOrder?
inOrder(nd->leftchild); 17
visit(nd);
inOrder(nd->rightchild); 9 27 6 16 20 31 12
Food for thought: which traversal will be useful for deep copy? Deep delete? 39
October 05, 2017
Cinda Heeren / Geoffrey Tien

16. Readings for this lesson
Koffman
Portions of Chapter 7.1 – 7.3 (for review)
Chapter 8.2 – 8.3 (Tree traversals and binary trees)
October 05, 2017
Cinda Heeren / Geoffrey Tien