After contacting the translator, it was found that the definition of interior points on page P660 is different from other domestic textbooks.
"Discrete Mathematics and its Applications (Original Book 8th Edition)" ISBN: 978-7-111-63687-8 The definition of interior points is shown in the figure below.
Therefore
"Discrete Mathematics and its Applications (Original Book 8th Edition)" ISBN: 978 -7-111-63687-8 Chapter 11 11.1.3 Properties of Trees
Theorem 3 on page 664 should be stated correctly.
The original theorem 3 is expressed as follows:
Theorem 3 A full m-ary tree with i interior points contains n=mi+1 vertices.
Example 1: Figure 1 is a full 3-way tree, as shown below:
Figure 1 is a full ternary tree, m=3, each branch point is connected to 3 nodes. The
interior points of the figure are marked in red, and
the number of interior points is i=13.
According to theorem 3, the vertices of the full ternary tree in Figure 1 are
n. =mi+1 = 3×13+1=40
Example 2: Figure 2 is a full 2-way tree, as shown below:
Figure 2 is a full 2-way tree m = 2, each branch point connects 2 nodes. The
interior points of the figure are marked in red, and the number of interior points is i = 7
. Theorem 3, the vertices of the full 2-way tree in Figure 2 are
n=mi+1 = 2×7+1=15
If you use the interior point definition of "Discrete Mathematics (4th Edition)" ISBN 978-7-302-61396-1, refer to 7.2 Root Tree and its Application 7.2.1 Root Tree and Its Classification P179
Vertices in the root tree with in-degree 1 and out-degree greater than 0 are called interior points.
In this case, the two books have different definitions of interior points. The formula needs to be adjusted to:
n=Total number of branch nodes ×i+1=m ×(i+1)+1
Figure 1 is expressed as:
m=3
i = 12
n = m(i+1)+1 = 3x(12+1)+1 = 40
Figure 2 is expressed as
m=2
i=6
n=m(i+1)+1 = 2x(6+1)+1 =15
This is recorded.