COP 3530
Assignment 8
Due: 20
th
Oct 2008
1. Consider a binary tree T, in which every non-leaf node has non-empty
left and right sub-trees. Assuming T has n leaf nodes, how many nodes in
total does it have? Justify your answer.
Hint: If we remove two leaf nodes that are under the same parent, every
non-leaf node of the new tree will still have non-empty left and right sub-
trees. How does this change the number of leaf (and non-leaf) nodes?
2. Consider an n-node complete binary tree T. Each node v of T is labeled
with a real number x
v
. You may assume that the real numbers labeling the
nodes are all distinct. A node v of T is a local minimum if the label x
v
is less
than the label x
w
for either child of v, or v's parent.
(a) Prove that any such tree has at least one local minimum.
(b) Write an efficient (fewest probes) algorithm for finding one local
minimum of T (maybe there are many, you only need to find one).
Calculate the time complexity for your algorithm.
3. Write a Java method
treeprint(node t, int n)
that will pretty-print a binary tree in indented format, using a preorder
traversal. The argument n is the current depth in the tree, whose value at
the beginning of the recursion is assumed to be zero. For example:
a (3)
... b (1)
... ... c(0)
... d (2)
... ... e (0)
... ... f (0)
... g (0)
In this tree, a has 3 children (b,d,g), b has one child (c), and d has 2 children
(e and f).
