CSCI 151 Fall 2012
Structural Induction Practice
A natural number n is either:
• 0, or
• n0 + 1, where n0 is a natural number.
Use this definition to connect the following algebraic expressions. The first three are done for you.
1. n2 to (n0 )2 :
n2 = (n − 1)2 + 2n − 1 = (n0 )2 + 2n − 1.
2. 2n4 to 2(n0 )4 :
2n4 = 2(n − 1)4 + 4n3 − 6n2 + 4n − 1 = 2(n0 )4 + 4n3 − 6n2 + 4n − 1.
0
0
3. 2n to 2n :
2n = 2 · 2n−1 = 2 · 2n .
0
4. 2n+1 to 2n +1 :
0
5. 2n + 1 to 2n + 1:
0
0
6. 2n+1 − 2n−1 + 1 to 2n +1 − 2n −1 + 1:
0
7. 3n+1 − 1 to 3n +1 − 1:
8. log2 (n) to log2 (n/2):
9. log2 (n) to log2 (2n):
10. n! to n0 !:
11. (n + 1)! to (n0 + 1)!:
12. (n + 1)! − 1 to (n0 + 1)! − 1:
Pn0
Pn
13.
i=0 i:
i=0 i to
Pn0 i
Pn i
14.
i=0 2 to
i=0 2 :
Pn i
Pn0 i
15.
i=0 3 to
i=0 3 :
Pn
Pn0
16.
i=0 (2i + 1) to
i=0 (2i + 1):
Pn
Pn0
17.
i=0 i · i! to
i=0 i · i!:
0
18. (1 + x)n to (1 + x)n (for x > −1):
19. 1 + n · x to 1 + n0 · x (for x > −1):
(continued on next page)
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A binary tree is either:
• empty, or
• a root node r with up to 2 binary subtrees L and R (and edges from r to its non-empty children).
Use this definition to connect the following expressions. The first few are done for you.
1. n(T ) to n(L), n(R), where n(T ) denotes the number of nodes of T :
2. h(T ) to h(L), h(R), where h(T ) denotes the height of T :
n(T ) = 1 + n(L) + n(R)
h(T ) = 1 + max{h(L), h(R)}
3. e(T ) to e(L), e(R), where e(T ) denotes the number of edges of T :
4. `(T ) to `(L), `(R), where `(T ) denotes the number of leaves of T :
5. n(d, T ) to n(?, L), n(?, R), where n(d, T ) denotes the number of nodes at depth d in T (the ? is
intentional; you need to figure out what depth parameter to pass in):
Recall that the heap-order property for a binary tree T states that:
for every node v of T , valueAt(v) ≤ valueAt(root of L) and valueAt(v) ≤ valueAt(root of R)
That is, every node in the tree has value less than that of each of its children.
Use this and the definition of a binary tree to connect the following expressions.
1. root(T ) to root(L), root(R), where root(T ) denotes the values of the root of T :
2. minV al(T ) to minV al(L), minV al(R), where minV al(T ) denotes the minimum value of all the
nodes in T :
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