Welcome.
To change this document, click the edit tab.
You can enter a number range you are interested in, like one to six:
^{
1
}
^{
2
(
2
)
}
^{
3
(
4
(
3
)
)
}
^{
4
(
3
)(
3
)
}
^{
5
(
6
(
6
(
5
)
)
)
}
^{
6
(
6
(
5
)
)(
5
)
}
After editing, click the "Expore" tab to see the results.
You can drag and drop circles to change the number-trees.
Double click a circle to re-root the tree with the selected number.
~~~
David W. Matula discovered that you can associate an integer with each rooted tree. Google "Matula Number" for more.
This site calculates the Matula number at each node of a given tree.
Notice how the leaves of Matula trees are prime numbers, and the forks are composite numbers.
For a given number of vertices (v) there are only so many unique trees that can be formed.
I enumerated trees up to seven vertices.
To inform this site to draw a Matula tree, I just enter a number into the edit pane.
This site generates all of the other vertices for you.
To re-root a tree, click on a number. To add a vertex, drag a circle from another tree. To get rid of a vertex, drag it out of the tree onto the page.
To enumerate a range of trees, use the format a:b where a and b are the starting and stoping number, in the edit tab.
~~~
Matula trees by number of vertices:
zero vertices -> one empty space
^{
0
}
one vertice -> one tree
^{
1
}
two vertices -> one tree
^{
2
(
2
)
}
three vertices -> one tree
^{
3
(
4
(
3
)
)
}
four vertices -> two trees
^{
5
(
6
(
6
(
5
)
)
)
}
^{
7
(
8
(
7
)(
7
)
)
}
five vertices -> three trees
^{
11
(
10
(
9
(
10
(
11
)
)
)
)
}
^{
17
(
14
(
12
(
13
)(
13
)
)
)
}
^{
53
(
32
(
53
)(
53
)(
53
)(
53
)
)
}
six vertices -> six trees
^{
31
(
22
(
15
(
15
(
22
(
31
)
)
)
)
)
}
^{
41
(
26
(
18
(
26
(
41
)
)(
23
)
)
)
}
^{
43
(
28
(
28
(
43
)(
43
)
)(
43
)
)
}
^{
59
(
34
(
21
(
20
(
29
)(
29
)
)
)
)
}
^{
67
(
38
(
24
(
37
)(
37
)(
37
)
)
)
}
^{
131
(
64
(
131
)(
131
)(
131
)(
131
)(
131
)
)
}
seven vertices -> eleven trees
^{
83
(
46
(
27
(
46
(
83
)
)(
46
(
83
)
)
)
)
}
^{
127
(
62
(
33
(
25
(
33
(
62
(
127
)
)
)
)
)
)
}
^{
131
(
64
(
131
)(
131
)(
131
)(
131
)(
131
)
)
}
^{
139
(
68
(
49
(
68
(
139
)(
139
)
)
)(
139
)
)
}
^{
157
(
74
(
36
(
74
(
157
)
)(
61
)(
61
)
)
)
}
^{
163
(
76
(
56
(
107
)(
107
)(
107
)
)(
163
)
)
}
^{
179
(
82
(
39
(
30
(
58
(
109
)
)(
47
)
)
)
)
}
^{
191
(
86
(
42
(
52
(
101
)(
101
)
)(
73
)
)
)
}
^{
241
(
106
(
48
(
89
)(
89
)(
89
)(
89
)
)
)
}
^{
277
(
118
(
51
(
35
(
44
(
79
)(
79
)
)
)
)
)
}
^{
331
(
134
(
57
(
40
(
71
)(
71
)(
71
)
)
)
)
}
It is easy to refer to trees and nodes with just a Matula number. No arbitrary labeling schemes are required (if you do not care to differentiate vertices that are equivalent under isomorphism).
Conjoining two trees together at vertex p and q is the same as multiplying p and q's Matula number.
Notice how composite numbers belonging to trees of vertex count v are limited to using prime numbers established at lower v trees? Can you prove this?
What other things can we say about the average number of primes as a function of v?
Trees can be entered as parenthesis, for example, two parenthesis sets next to each other is equal to
^{
4
(
3
)(
3
)
}
.
If you want to know what the nth prime number is, just surround n with parenthesis like this: (n). Here is the hundreth prime:
^{
541
(
200
(
519
(
2062
(
8221
)
)
)(
519
(
2062
(
8221
)
)
)(
541
)(
541
)
)
}
.
Have Fun
-- William