RECURSION:
In programming, recursion is a call to the same method from a method. Recursion is a good
problem solving approach
Example1: write a function that computes the sum of numbers from 1 to n
1. Use a loop.
2. Recursively.
1-int sum (int n)
{
int s = 0;
for (int i=0; i<n; i++)
s+= i;
return s;
}
2-int sum (int n)
{
int s;
if (n == 0)
return 0;
else
s = n + sum(n-1);
return s;
}
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Example2: Factorial (N!)
• N! = (N-1)! * N [for N > 1]
• 1! = 1
• 3! = 2! * 3= (1! * 2) * 3= 1 * 2 * 3
• Recursive design: - Decomposition: (N-1)! - Composition: * N - Base case: 1!
public static int factorial(int n)
{
int fact;
if (n > 1) // recursive case (decomposition)
fact = factorial(n – 1) * n; // composition
else // base case
fact = 1;
return fact;
}
public static intfactorial(int 3)
{
int fact;
if (n > 1)
fact = factorial(2) * 3;
else
fact = 1;
public static int factorial(int 2)
return fact;
{
}
int fact;
if (n > 1)
fact = factorial(1) * 2;
else
fact = 1;
public static int factorial(int 1)
return fact;
{
}
int fact;
if (n > 1)
fact = factorial(n - 1) * n;
else
fact = 1;
return fact;
}
H.W: write arecursive function for:
1- gcd(m,n)
2
2- power(2^n)
Binary Search Trees
A Binary Search Tree is a binary tree with the following properties:
 All items in the left subtree are less than the root.
 All items in the right subtree are greater or equal to the root.
 Each subtree is itself a binary search tree.
Basic Concepts
Binary search trees provide an excellent structure for searching a list and at the same time for
inserting and deleting data into the list.
Valid Binary Search Trees
(a), (b) - complete and balanced trees;
(d) – nearly complete and balanced tree;
(c), (e) – neither complete nor balanced trees
Invalid Binary Search Trees
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Trees: basic definitions and terminology
Contrary to arrays, stacks, queues and sequences all of which are one-dimensional data
structures, trees are two-dimensional data structures with hierarchical relationship between
data items.
Definition 1 A tree is a non-empty collection of vertices (nodes) and edges that satisfy certain
requirements.
Definition 2 A path in a tree is a list of distinct vertices in which successive vertices are
connected by edges in the tree.
One node in the tree is designated as the root. Each tree has exactly one path between the root
and each of the other nodes. If there is more than one path between the root and some node, or
no path at all, we have a graph.
Definition 3 A set of trees is called a forest.
Definition 4 (Recursive definition) A tree is either a single node or a root node connected to a
forest.
Example of a tree:
More definitions
Definition 5 An ordered tree is a tree in which the order of children is specified.
Definition 6 A level (depth) of a node is the number of nodes on the path from that node to the
root.
Definition 7 The height (maximum distance) of a tree is the maximum level among all of the
nodes in the tree.
Definition 8 The path length of a tree is the sum of the levels of all the nodes in the tree.
Definition 9 A tree where each node has a specific number of children appearing in a specific
order is call a multiway tree. The simplest type of a multiway tree is the binary tree. Each node
in a binary tree has exactly two children one of which is designated as a left child, and the other
is designated as a right child.
Definition 10 (Recursive definition) A binary tree is either an external node, or an internal node
and two binary trees.
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binary trees examples
1. Binary tree for representing arithmetic expressions. The underlying hierarchical relationship
is that of an arithmetic operator and its two operands.
Arithmetic expression in an infix form: (A - B) + C * (E / F)
Note that a post-order traversal of this tree (i.e. visiting the left subtree first, right subtree
next, and finally the root) returns the postfix form of the arithmetic expression, while the preorder traversal (root is visited first, then the left subtree, then the right subtree) returns the
prefix form of the arithmetic expression.
BST Operations
We discuss four basic BST operations: traversal, search, insert, and delete; and develop
algorithms for searches, insertion, and deletion.
• Traversals
• Searches
• Insertion
• Deletion
Binary Tree Traversal
To traverse a tree means to visit all the nodes in some specified order. For example, you might
visit all the nodes in order of ascending key value. Thus the process is most easily described
through recursion.
The names given for particular style of traversal came from the position of root element with
regard to the left and right nodes. Imagine that the left and right nodes are constant in space, then
the root node could be placed to the left of the left node (pre-order), between the left and right
node (in-order), or to the right of the right node (post-order).
Depth-first traversal
To traverse a non-empty binary tree in preorder, perform the following operations recursively at
each node, starting with the root node:
1. Visit the root.
2. Traverse the left subtree.
3. Traverse the right subtree.
To traverse a non-empty binary tree in inorder (symmetric), perform the following operations
recursively at each node:
1. Traverse the left subtree.
2. Visit the root.
3. Traverse the right subtree.
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To traverse a non-empty binary tree in postorder, perform the following operations recursively
at each node:
1. Traverse the left subtree.
2. Traverse the right subtree.
3. Visit the root.
Breadth-first traversal
Trees can also be traversed in level-order, where we visit every node on a level before going to a
lower level. This is also called Breadth-first traversal.
Depth-first
As an example the tree above , the traversals are
 Preorder traversal sequence: F, B, A, D, C, E, G, I, H (root, left, right)
 Inorder traversal sequence: A, B, C, D, E, F, G, H, I (left, root, right); note how this
produces a sorted sequence
 Postorder traversal sequence: A, C, E, D, B, H, I, G, F (left, right, root)
Breadth-first

Level-order traversal sequence: F, B, G, A, D, I, C, E, H
Example 1
Inorder a, b, c, d, e, f, g, h, i, j, k
Preorder f, b, a, d, c, e, i, h, g, k, j
Postorder a, c, e, d, b, g, h, j, k, i, f
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Example 2
Inorder Traversal (LrR)
The result is D, B, E, A, F, C, G
Postorder Traversal (LRr)
The result is D, E, B, F, G, C, A
Preorder Traversal (rLR)
The result is A, B, D, E, C, F, G
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Example 3
A preorder traversal of the tree shown in figure above visits the nodes in the following order:
A, B, C, D, E, F, G, H, I
A postorder traversal of the tree shown in figure above visits the nodes in the following order:
C, B, F, G, E, I, H, D, A
An inorder traversal of the tree shown in figure above visits the nodes in the following order:
B, C, A, F, E, G, D, I, H
Example 4
Preorder Traversal 23, 18, 12, 20, 44, 35, 52
Postorder Traversal 12, 20, 18, 35, 52, 44, 23
Inorder Traversal 12, 18, 20, 23, 35, 44, 52 Inorder traversal of a binary search tree
produces a sequenced list
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