Lesson 16: Recursion

1
Lesson 16 Recursion

• Objectives
• Explain why recursively defined functions are
  well-defined
• Describe how sets and structures can be
  recursively defined
• Tell the importance of recursive functions
• Explain how a set of rooted trees and the sets of
  extended binary trees are defined

• Outline
• Recursive functions
• Defining Sets and Structures Recursively
• Trees
• Reading 3.4

2
Recursively defined sequences (functions)

• Define base case (at 0, 1, etc)
• Define a rule for calculating next number in
  sequence, given previous number.

3
Recursively defined functions

• f(n) an

4
Inferred Sequence

• 0, 2, 6, 12, 20, 30, 42, ...
• Imply a recursive definition

5
Fibonacci Sequence

• Fib(n) Fib(n-1) Fib(n-2)
• http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
  acci/fibnat.html

6
Recursively Defined Sets

• Basis 2 ? S
• Recursive step If x ? S and y ? S, then xy ? S
• What are the elements of S?

7
Recursively defined Sets

• S 4, 7, 10, 13, 16, 19 ...
• What is a rule base for defining S? (in class)

8
Recursively defining Rooted Trees

• Basis Step A single root is a rooted tree
• Recursive step Taking a rooted Tree T and
  connecting it to the existing rooted tree at any
  vertex produces a rooted tree.

9
Extended Binary Tree

• Basis Step The empty set is an extended binary
  tree
• Recursive step If T1 and T2 are EBTs, connect a
  root to the root of the left subtree of T1 and to
  the root of the right subtree of t2 this
  produces a new EBT

10
Full Binary Tree

• Basis Step A single vertex r
• Recursive Step Connect two FBTs to root.

11
Variables

• Consider the alphabets
• a 0-9, a-z, A-Z
• b a-z
• Define a recursive method of generating valid
  variable names.