CSE 260
Homework 9- Recursive Definitions
Ignore the proofs by mathematical induction part of the problems.
1. Section 4.3: 20
2. Prove that 5n−4n−1 is divisible by 16 for nZ +
3. Recursively define b0=b1=b2= 1 and bn=bn−1+bn−3for n≥3.
(a) Calculate bnfor n= 3,4,5
(b) Show by using the second principle (strong) of mathematical induc-
tion that
bn≥2bn−2for n≥3
4. Prove by mathematical induction
13+ 23+... +n3= (1 + 2 + ... +n)2
5. The set of rooted trees, where a rooted tree consists of a set of vertices
containing a distinguished vertex called the root, and edges connecting
these vertices, can be defined recursively by the following steps:
(a) Basis Step: A single vertex is a rooted tree.
(b) D Recursive Step: Suppose that T1, T2, ...Tnare disjoint rooted
trees with roots r1, r2, ..., rnrespectively. Then the graph formed
by starting with a root r, which is not in any of the rooted trees
T1, T2, ...Tn, and adding an edge from r to each of the vertices
r1, r2, ..., rn, is also a rooted tree.
Construct the following tree using the above definition. Show all
the steps of your construction.
6. Followiing is a recursive function defined for a binary tree, T, that returns
the height of the tree. Give the sequence of values of the variable val
that the function generates when the binary tree, T, is the one given in
problem 3 above.
