Nt1310 Unit 1 Case Study

1. How many ancestors does a node at level n in a binary tree have? Provide justification.

The definition of a binary tree states that if a tree is not empty, then a root node has two sub trees Tr and Tl, such that Tr and Tl are binary trees. Under this definition, every node, except for the root node, has one parent. Levels of the tree is a measure of distance from the root node, assuming the root node's level is 1, node n's level is 1 plus the level of its parent. Since the root node is level 1 and every node has one parent, the number of ancestors of a node at level n is n-1.

2. Prove that a strictly binary tree (regular binary tree) with n leaves contains 2n-1 nodes. Provide justification.

A strictly binary tree, full binary tree requires

all nodes to have either 2 children or 0 children. A leaf node contains node children and since we are dealing with binary trees exactly one parent. To satisfy a full binary tree require, a leaf node’s grandparent must have two children, thus for every leaf node there are 2n-1 nodes in the tree (the root node has no parent).

3. How many binary trees exist with n nodes and level k = 3? Justify your answer. Do not count isomorphic tree (ones with the same physical structure). See the diagram in the supplement.

The maximum level for a tree with n nodes n-1, so k is at nost n-1
N=4
N=4, k=[0,3], trees=3

Trees
N=4, K=0
K=1
K=2
K=3
3 trees we will have.

4. Explain in detail that if m pointer fields are set aside in each node of a general m-ary tree to point to a maximum of m child nodes, and if the number of nodes in the tree is n, the number of null child pointer fields is n*(m-1)+1.

M-ary trees have MN pointers (m pointer fields for each node). There are N-1 edges in a M-ary tree (non-Null). Therefore, the number of null child pointer fields is: MN-(n-1)=MN – N+1=N(M-1)+1.

5. Implement maketree, setleft, and setright for right in-threaded binary trees using the sequential rayay representation.

class node {
    int text;
    node left;
    node right;
public class Thread{
    static node maketree(int x) {
        node p;
        p = getnode();
        p.text = x;
        p.left = null;
        p.right = null;
        p.rthread = true;
        return p;
    }
    static node getnode() {
        return new node();
    }
    static void setleft(node p, int x) {
        node q;
        if (p == null)
        else {
            q = getnode();
            q.text = x;
            p.left = q;
            q.left = null;
            q.right = p;
            q.rthread = true;
}
    static void setright(node p, int x) {
        node q, r;
        if (p == null)
        else {
            q = getnode();
            q.text = x;
            r = p.right;
            p.right = q;
            p.rthread = false;
            q.left = null;
            q.right = r;
            q.rthread = true;
        }
6.

Implement inorder traversal for the right in-thread tree in the previous problem.

 void sort(int ray[], int start, int end)

{

if(start > end)

    return;

sort(ray, start*2 + 1, end);

printf("%d ", ray[start]);

sort(ray, start*2 + 2, end);

}

int main()

{

int raySize = sizeof(ray)/sizeof(int);

sort(ray, 0, ray_size-1);

getchar();

return 0;

}

7. Define the Fibonacci binary tree of order n as follows: If n=0 or n=1, the tree consists of a single node. If n>1, the tree consists of a root, with the Fibonacci tree of order n-1 as the left subtree and the Fibonacci tree of order n-2 as the right subtree.  Write a method that builds a Fibonacci binary tree of order n and returns a pointer to it.

struct node {

   int text;

   struct node *left;

   struct node *right;

}

typedef struct node *node;

node acci(int n)

{

   node p = getnode();

   if (n == 0 || n == 1) {

      p->left = NULL;

      p->right = NULL;

   }

   else

   {

      p->left = acci(n - 1);

      p->right = acci(n - 2);

   }

   return p;

}

8. Answer the following questions about Fibonacci binary tree defined in the previous

problem.

 Biometrics are about the same with a 1-bit difference and the hash will not match.

a) Is such a tree strictly binary?

Yes, the Fibonacci binary tree is strictly binary.

b) What is the number of leaves in the Fibonacci tree of order n?

A Fibonacci tree consist of n-1 internal nodes (each having two Childs) and n terminal nodes or leaves.

c) What is the depth of the Fibonacci tree of order n?

Depth is log(n).