Recursion is a programming technique in which a function calls itself directly or indirectly. Using a recursive algorithm, certain problems can be solved quite easily. Examples of such problems are Towers of Hanoi (TOH), Inorder/Preorder/Postorder Tree Traversals, DFS of Graph, etc.

1. Week: 14 Recursion & its
Application
Batch 9 Fall 2021

2. Contents
 Recursion concept
 Example
 Tower of Hanoi using Recursion method

3. Recursion
 The process in which a function calls itself is called recursion. The function calling itself is called
recursive function
 Let us assume a programmer has to find sum of first n natural numbers
 There are two approaches to find the sum of series
Iterative Algorithm: (using loops for repetitions)
𝑆𝑛 = 1 + 2 + 3 + … + 𝑛
Recursive Algorithm: (using function for
repetitions)
𝑓 𝑛 = 0 𝑛 = 0
𝑓 𝑛 = 𝑛 + 𝑓 𝑛 − 1 𝑛 ≥ 1

5. Example 2:
 Find factorial of a number entered by the user.
Iterative Algorithm: (using loops for repetitions)
𝐹𝑛 = 𝑛 𝑛 − 1 𝑛 − 2 𝑛 − 3 … .1
Recursive Algorithm: (using function for
repetitions)
𝑓 𝑛 = 1 𝑛 = 1
𝑓 𝑛 = 𝑛 ∗ 𝑓 𝑛 − 1 𝑛 > 1

7. Recursion Application
 Tower of Hanoi
 Tree Traversal (In order, preorder, post order)
 Depth first search

8. Tower of Hanoi – Iterative Solution

9. Tower of Hanoi – Iterative Solution
1. Calculate the total number of moves required i.e. "pow(2, n) - 1" here n is number of disks.
2. If number of disks (i.e. n) is even then interchange destination pole and auxiliary pole.
3. for i = 1 to total number of moves:
if i%3 == 1:
legal movement of top disk between source pole and destination pole
Either there will be minimum flow or least flow btw poles
if i%3 == 2:
legal movement of top disk between source pole and auxiliary pole
Either there will be minimum flow or least flow btw poles
if i%3 == 0:
legal movement of top disk between auxiliary pole and destination pole
Either there will be minimum flow or least flow btw poles

10. Tower of Hanoi – Recursive Solution
 𝑇𝑂𝐻 𝑛, 𝑠, 𝑑, 𝑎 =
𝑀𝑜𝑣𝑒 𝑑𝑖𝑠𝑘 1 𝑓𝑟𝑜𝑚 𝑠 𝑡𝑜 𝑑 ; 𝑛 = 1
𝑇𝑂𝐻 𝑛 − 1, 𝑠, 𝑎, 𝑑
𝑀𝑜𝑣𝑒 𝑑𝑖𝑠𝑘 𝑛 𝑓𝑟𝑜𝑚 𝑠 𝑡𝑜 𝑑 ; 𝑛 > 1
𝑇𝑂𝐻 𝑛 − 1, 𝑎, 𝑑, 𝑠