"Discrete Mathematics and Its Applications (8th Edition of the Original Book)" ISBN978-7-111-63687-8 Chapter 11 11.1.3 The extension of Theorem 3 on page 664 of the Properties of Trees section
Theorem 3: An m-ary tree with i interior points contains n=mi+1 vertices.
See the discussion on different point definitions in my blog post.
If for an m-ary regular tree, that is, there are exactly m sons of any branch node, how should the formula be expressed?
The figure below draws a 5-way regular tree, as shown below:
According to the definition of interior point in "Discrete Mathematics (4th Edition)" ISBN 978-7-302-61396-1:
It can still be based on the formula:
n=m(i +1) +1, n represents the number of vertices, i represents the number of interior points, and the
calculation
m=5
i=3
n=m(i+1)+1 = 5x(3+1)+1 = 21
meets the requirements.
Example 9 on page 664 of "Discrete Mathematics and Its Applications (8th Edition of the Original Book)": Example 9
: Suppose someone sends a chain letter. Ask each person who receives the letter to send it to 4 other people. Some people did so, but others did not send the letter. The chain of letters ends if no one receives more than one letter, and if more than 100 people have read the letter but not sent it, then Including the first person, how many people read the letter? How many people have sent letters?
Solution: This is a 4-way regular tree problem.
Define a chain letter with a 4-way regular tree.
The number of leaves: l = 100
m=4
i expresses the number of interior points
according to the following two formulas:
Formula 1: n=m(i+1)+1
Formula 2: n=i+1+l (number of interior points + number of roots + number of leaves)
Bringing in
n=4(i+1)+1 = i+1+100
gives us
i=32
n=133
. Therefore, including the first person (the root of the graph), there are 133 people who have read the letter, and there are 32+ 1=33 people have sent the letter.