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Question 4:  2d-array, nested for-loops

Suppose someone has measured the elevations at a soccer field as shown in 
following picture.  At all the crosses a measurement of the elevation was 
taken.  The distances between the measurements in the x-direction 
(dx) and y-direction (dy) are 25m and 20m, respectively.  The surveyer knew 
about Matlab and stored the measurements in a 5-by-6 matrix that he saved 
in a Matlab file called elevat.mat.

   
     +------+------+------+------+------+
     |      |      |      |      |      |
     |      |      |      |      |      |
     +------+------+------+------+------+
     |      |      |      |      |      |
     |      |      |      |      |      |     Y
     +------+------+------+------+------+     ^
     |      |      |      |      |      |     |
     |      |      |      |      |      |     |
     +------+------+------+------+------+     -----> X
     |      |      |      |      |      |
     |      |      |      |      |      |
     +------+------+------+------+------+


You are asked to write a Matlab script file to determine the gradients 
in the x and y directions at all interior points (3*4=12 points), i.e., 
all points excluding the points on the boundary.  The gradients should be 
stored in two 3 by 4 matrices and written to files called grad_x.mat and 
grad_y.mat.

This is how you determine the gradient for an interior point where i
refers to the row and j to the column of matrix elevat, respectively:

     grad_x(i,j)=(elevat(i,j+1)-elevat(i,j-1))/(2*dx)   
     grad_y(i,j)=(elevat(i+1,j)-elevat(i-1,j))/(2*dy)

Which values of the matrix elevat did you not use for your
calculations?


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Question 5:  recursion

Write functions to determine the following:

a) N!  (N factorial)

b) the sum of N numbers in the Fibonnaci sequence: 1 2 3 5 8 13 ...

c) the Nth number in the Fibonnaci sequence.  E.g., the 5th number is 8.


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Question 6:  more recursion

Computer scientists refer to binary trees as a structure that can be
illustrated pictorially by the following example:

            A
           / \
          B   C
         /
        D
       / \
      X   Y

Here, A is the root of the tree and B and C are its children.  C has no 
children and B has one child D, and so on.  A, B, C, etc. are called nodes.
In a binary tree, a node can have zero, one, or two children.  A node that 
has no children is called a leaf.  E.g., X is a leaf.  In Matlab, a binary
tree can be represented using cell-arrays.  The example above can be 
represented as

   {'A', {'B', {'D',{'X', {}, {}}, {'Y', {}, {}}}, {}}, {'C', {}, {}}}

The idea is that the binary tree with root A is given by
   {'A', left-subtree, right-subtree}
where left-subtree in this case is a tree with root B and therefore it will
be represented by {'B', ..., ...}.  C here does not have any children, so we
say C has empty left-subtree and empty right-subtree reprsented by {}.

Write a function $fringe$ that returns an array of the leaves of a given tree.
E.g., if the above array is the input to function $fringe$, the result should
be {'X', 'Y', 'C'}.

Strategy:  recursion on the tree


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