Question

In: Advanced Math

Prove that a full m-ary tree of height h has at least h(m − 1) leaves.

Expert Solution

The correct statement is: a full -ary tree of height has at least

We use induction on .

If , then a full -ary tree of height consists of a root and children all of whom are leaves. Thus, the number of leaves is .

Suppose that for some integer it is true that every full -ary tree of height at most has at least leaves. Consider a full -ary tree of height , and consider the subtree obtained by deleting all the leaves of which are at height . Since we are not deleting any of the internal nodes of or any leave at height , the resulting subtree is full -ary tree of height , and by induction hypothesis, this resulting subtree has at least leaves. In some of these are internal nodes (because the leaves at height , of which there is at least one, are connected to some of these nodes), hence, we get at least

leaves. Thus, the number of leaves is at least . Since , we get .

Thus, we have proved, by induction, that a full -ary tree of height has at least leaves.

Colby Messinger answered 2 years ago

As a quarterback throws the football, it leaves his hand at a height of h and...

As a quarterback throws the football, it leaves his hand at a height of h and at an initial speed of v0 at an angle θ above the horizontal. The receiver can’t get to the football in time and it falls onto the field. What is the ball’s speed as it hits the ground? Solve this problem in two ways: (a) using kinematics, explicitly calculating the trajectory the ball takes (b) using conservation of (kinetic + potential) energy

Let X represent the full height of a certain species of tree. Assume that X has...

Let X represent the full height of a certain species of tree. Assume that X has a normal probability distribution with a mean of 77 ft and a standard deviation of 3.5 ft. A tree of this type grows in my backyard, and it stands 65.8 feet tall. Find the probability that the height of a randomly selected tree is as tall as mine or shorter. P(X<65.8)= My neighbor also has a tree of this type growing in her backyard,...

Prove that a binary tree that is not full cannot correspond to an optimal prefix code....

Prove that a binary tree that is not full cannot correspond to an optimal prefix code. The proof should first consider a prefix-free code C, whose corresponding binary tree T has some node with only one child; show that one can transform T into another binary tree T', whose corresponding code C' has smaller average length and so is better than C. In your proof you need to indicate the transformation from T into T' and explain why the code...

A basketball leaves a player's hands at a height of 2.25 m above the floor. The...

A basketball leaves a player's hands at a height of 2.25 m above the floor. The basket is 3.05 m above the floor. The player likes to shoot the ball at a 36.0 ∘ angle If the shot is made from a horizontal distance of 12.00 m and must be accurate to ±0.32 m (horizontally), what is the range of initial speeds allowed to make the basket? Enter your answers numerically separated by a comma.

1. Ablockofmass M = 4kg, cross-sectional area A = 1 m^2, and height h = 60...

1. Ablockofmass M = 4kg, cross-sectional area A = 1 m^2, and height h = 60 cm sits in a liquid with density ρ = 12 kg/m^3. a) Find the depth d the block will sit in the water when in equilibrium. b) At time t=0, the block is released from being completely submerged in the liquid, calculate the amplitude and frequency of oscillation.

DEPTH counting starts at 1 at the root. How many total nodes does a 7-ary tree...

DEPTH counting starts at 1 at the root. How many total nodes does a 7-ary tree have if its depth is 8? Hash the given file (6,22,9,14,13,1,8) and use linear probing and the table size of 11. How many total collisions occurred? What is the loading density of the table?

Let A be an m x n matrix. Prove that Ax = b has at least...

Let A be an m x n matrix. Prove that Ax = b has at least one solution for any b if and only if A has linearly independent rows. Let V be a vector space with dimension 3, and let V = span(u, v, w). Prove that u, v, w are linearly independent (in other words, you are being asked to show that u, v, w form a basis for V)

a) A full conical water tank has height 3 m and diameter across the top 2...

a) A full conical water tank has height 3 m and diameter across the top 2 m. It is leaking water at a rate of 10,000 cm^3 per minute. How fast is the height of the water decreasing when the volume of water is 3,000,000 cm^3 ? b) Two planes are approaching the same air traffic control tower at equal and constant altitude. Plane A is heading due north at 200 mph, while Plane B is heading due west at...

A person 100 meters from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees. Estimate the height h of the tree to the nearest tenth of a meter.

Use proper calculus. (a) A conical shell has mass M, height h, and base radius R,...

Use proper calculus. (a) A conical shell has mass M, height h, and base radius R, Assume it is made from thin sheet of uniform thickness, Derive its center of mass and moment of inertia about its symmetry axis. (b) Derive the formula for the center of mass and the moment of inertia of a solid sphere, and then that of the moment of inertia about an axis tangent to the surface. (c) Derive the formula for the moment of...