Two graphs with distinct vertices


The idea is to build "trees" with distinct positive integers k at every fork – k being the exact number of paths leading k to other integers (example above:  there are 3 paths leading "3" to 3 other integers and 4 paths leading "4" to 4 other integers). 

 We don't accept "crossroads" in the trees presented above and below: when you go from one integer to another, you don't have any choice, your paths leads you to that integer and nothing else. 

The "tree" above is called (by me) the Maxi-tree.
This is because no paths leads a(n) to an integer < a(n), except exactly one ("7" for instance can lead you to 4, 18, 19, 20, 21, 22, 23 but not to 1, 2, 3, 5 or 6; we see indeed that 4 is the only accessible integer < 7). We have here:
1 linked to 2, 
2 linked to 1 and 3, 
3 inked to 2, 4 and 5,
4 linked to 3, 6, 7 and 8,
5 linked to 3, 9, 10, 11 and 12,
6 linked to 4, 13, 14, 15, 16 and 17,
7 linked to 4, 18, 19, 20, 21, 22 and 23,
8 linked to 4, 24, 25, 26, 27, 28, 29 and 30,
9 linked to 5, 31 to 38,
10 linked to 5, 39 to 47,... etc.
The bold/pink terms here form the sequence 2, 3, 5, 8, 12, 17, 23, 30, 38, 47,... which is the triangular numbers +2.
                                                             
The second "tree" (hereunder) is the Mini-tree – it was progressively built linking "1" to 1 integer ("2"), linking "2" to 2 integers ("1" and "3"), linking "3" to 3 integers ("2", "4" and "5"), linking "4" to 4 integers ("3", "5", "6" and "7") , linking "5" to 5 integers ("3", "4", "6", "7", "8"),... linking "k" to k integers – the two constraints being:
1) "k" has links with as many integers < k as possible;
2) any integer > k must be the smallest available one not yet present in the tree.
(click the picture to enlarge it)
(doesn't this picture remind you of the Three utilities problem?) Anyway, we have here:
1 linked to 2, 
2 linked to 1 and 3, 
3 inked to 2, 4 and 5,
4 linked to 3, 5, 6 and 7,
5 linked to 3, 4, 6, 7 and 8,
6 linked to 4, 5, 7, 8, 9 and 10,
7 linked to 4, 5, 6, 8, 10, 11 and 12,
8 linked to 5, 6, 7, 9, 12, 13, 14 and 15,
9 linked to 6, 8, 10, 15, 16, 17, 18, 19 and 20,
10 linked to 6, 7, 9, 11, 25, 26, 27, 28, 29 and 30, 
11 linked to 7, 10, 12, 25, 26, 27, 28, 29, 30, 31 and 32,
12 linked to 7, 8, 11, 13, 32, 33, 34, 35, 36, 37, 38 and 39,
13 linked to 8, 12, 14, 39, 40, 41, 42, 43, 44, 45, 46, 47 and 48,
14 linked to 8, 13, 15, 48 to 58,
15 linked to 8, 9, 14, 16, 58 to 68,
16 linked to 9, 15, 17, 68, 69 to 80, ... etc.

The Mini-tree gives us the "bold and pink" sequence 2, 3, 5, 7, 8, 10, 12, 15, 20, 30, 32, 39, 48, 58, 68, 80, ... which is not in the OEIS.
The leftmost branch L of the tree doesn't seem to be either in the database:
L = 1, 2, 3, 4, 6, 9, 16, 69, ...

 And what about growing prime mini-trees?
Best,
É.

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