Question

Trees with the most leaves.

(a) If T is a tree with n vertices, what is the most leaves that it can have? Your answer will be an expression involving the variable n. Explain your reasoning. Be sure to address small values for n (e.g., n = 1 or 2).

(b) Draw a tree with eight vertices that has the most number of leaves possible.

Answer #1

How many trees T are there on the set of vertices
{1, 2, 3, 4, 5, 6, 7} in which the vertices 2 and 3 have degree 3,
vertex
5 has degree 2, and hence all others have degree 1? Do not
just
draw pictures but consider the possible Pr¨ufer codes of these
trees.

Let f : [n] ? [n] be a function and let Tf be a labeled tree on
n vertices, constructed from f using the procedure demonstrated in
class. Suppose that f takes exactly k different values. Show that
Tf has at most n?k vertices with degree at least four.
Hint: First, determine the maximum number of leaves in Tf .

A tree T has 8 vertices, at least two of which have degree
3.
a) How many edges are there?
b) What are the possible vertex degrees for T in non–increasing
order?
c) What are the possible forms for T up to isomorphism?
The answer to part a is "7" while the answer to part be is "
3,3,3,1,1,1,1,1 and 3,3,2,2,1,1,1,1"
There are six possible answers for part c. how? and what are the
answers?

You have been given the order of the nodes (vertices) as visited
by a postorder traversal and also the order of the nodes (vertices)
as visited by an inorder traversal, do you have enough information
to reconstruct the original tree? (We assume that the nodes all
have unique values).
If your answer is "yes" can you explain it in an example tree of
about 7 to 10 nodes and see what happens. If your answer is "no",
explain that too.

A simple undirected graph consists of n vertices in a single
component.
What is the maximum possible number of edges it could have?
What is the minimum possible number of edges it could have?
Prove that your answers are correct

Let T be a complete binary tree such that node v stores
the entry (p(v), 0), where p(v) is the level number of v. Is tree T
a heap? Why or why not?
I know that a complete binary tree is a heap, but shouldn't we
also take into consideration the values that it is storing into the
tree: (p(v), 0)? The heap tree could be either a min-heap or
max-heap. If we order the the value based of p(v)...

9. (Extra Credit.)
A small sample - the T-distribution) The trees around the
Quincy College parking lot have an average length of 4 inches. The
leaf lengths are normally distributed, coming from a
population with a standard deviation of 0.6 inches. The
trees around the Quincy College parking lot are subject to the
stimulating aroma of MBTA exhausts. A sample of 20 leaves was taken
from these trees and the average leaf length was found to be 4.2
inches. With...

Suppose you have a spinner comprised of six equal spaces: a
orange 1, a yellow 2, a green 3, a pink 4, a blue 5, and a purple
6. Consider an experiment that consists of randomly spinning the
spinner twice.
Draw a tree diagram that shows all of the possible outcomes of
this experiment. That is, draw a tree diagram that illustrates all
possible two spin outcomes.
Count the number of paths that yield anything but
green or pink on...

The special case of the gamma distribution in which α is a
positive integer n is called an Erlang distribution. If we replace
β by 1 λ in the expression below, f(x; α, β) = 1 βαΓ(α) xα − 1e−x/β
x ≥ 0 0 otherwise the Erlang pdf is as follows. f(x; λ, n) = λ(λx)n
− 1e−λx (n − 1)! x ≥ 0 0 x < 0 It can be shown that if the times
between successive events are...

Suppose we have a binomial distribution with n trials
and probability of success p. The random variable
r is the number of successes in the n trials, and
the random variable representing the proportion of successes is
p̂ = r/n.
(a) n = 44; p = 0.53; Compute P(0.30
≤ p̂ ≤ 0.45). (Round your answer to four decimal
places.)
(b) n = 36; p = 0.29; Compute the probability
that p̂ will exceed 0.35. (Round your answer to four...

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