### Matula Number Explorer

#### by William Sharkey

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