Recursion and Binary Tree

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Recursion and Binary Tree

• ICS 51 Introductory Computer Organization

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Recursion

• Recursion in computer programming is exemplified
  when a function is defined in terms of itself.
• Advantage of recursion Simple to write and
  understand.
• Disadvantage Amount of memory required is
  proportional to the depth of recursion

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Problem 1

• Implement the Factorial function using assembly.
• Given an unsigned integer number n,we define
  Factorial(n) as
• Factorial(n) 1 , if n0 or n1
• n Factorial(n-1), if ngt1

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Binary tree and binary search tree

• A binary tree is a tree data structure in which
  each node has at most two children.
• Typically the child nodes are called left and
  right.
• A binary search tree (BST) is a binary tree data
  structure which has the following properties
• Each node has a value.
• The left subtree of a node contains only values
  less than the node's value.
• The right subtree of a node contains only values
  greater than or equal to the node's value.

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An example
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Searching

• Performed recursively because of the order in
  which values are stored.
• Begin by examining the root. If the value we are
  searching for equals the root, the value exists
  in the tree.
• If it is less than the root, then it must be in
  the left subtree, so we recursively search the
  left subtree in the same manner.
• Similarly, if it is greater than the root, then
  it must be in the right subtree, so we
  recursively search the right subtree.
• If we reach a leaf and have not found the value,
  then the item is not where it would be if it were
  present, so it does not lie in the tree at all.

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Insertion

• Insertion begins as a search would begin
• If the root is not equal to the value, we search
  the left or right subtrees as before.
• Eventually, we will reach an external node and
  add the value as its right or left child,
  depending on the node's value.
• In other words, we examine the root and
  recursively insert the new node to the left
  subtree if the new value is less than to the
  root, or the right subtree if the new value is
  greater or equal than the root.

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Example Add 9 and 2
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2
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Tree Traversal

• Starting at the root of a binary tree, there are
  three main steps that can be performed with the
  order that they are performed defining the
  traversal type.
• performing an action on the current node
  (referred to as printing the node)
• repeating the process with the left subtree
• repeating the process with the right subtree

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In order traversal

• To traverse a non-empty binary tree in inorder,
  perform the following operations
• Traverse the left subtree in inorder
• Print the root of current subtree.
• Traverse the right subtree in inorder.
• inorder(node)
• if node.left ? null then
• inorder(node.left)
• print node.value
• if node.right ? null then inorder(node.righ
  t)

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Example In order
Remember - Left - Print - Right
In-order (LPR) traversal yields A, B, C, D, E,
F, G, H, I
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Problem 2

• Given an array of integers, build a binary search
  tree containing the elements of the array.
• You have to write a recursive function
  insertNode(struct tNode root,struct tNode
  newNode, unsigned long data), where root is the
  root of the binary search tree.
• After building the tree , you will do an inorder
  (left subtree-root-right subtree) traversal using
  the sortedTree(struct tNode root, unsigned long
  a , unsigned long counter) function and you
  will store the elements of the tree in the a
  array. After doing this, the a array will
  contain the elements of the tree sorted in
  ascending order.
• Next, you have to implement the
  reversedTree(struct tNode root, unsigned long a
  , unsigned long counter) function, which will
  place the elements of the tree into the a array ,
  sorted in descending order (a right
  subtree-root-left subtree traversal)