User blog:Primussupremus/Slow growing trees

You start off with one node or point you then branch off twice meaning you now have 3 nodes this completes 2 steps. Next you branch off 3 nodes from the 2 nodes you branched off previously making 9 nodes in total and 6 on the 3rd level. After that you branch off 4 nodes from each of the six previous nodes  meaning you will have 24 nodes on  the 4th level and 33 nodes in total. You then branch of 5 nodes from each of the 24 making 120 nodes on the 5th level and 153 nodes in total. You branch off 6 nodes from each of the 120 nodes making 720 nodes for the 6th level and 873 nodes in total. After that you branch off 7 nodes from each of the 720 nodes making 5040 nodes for the 7th level and 5913 nodes in total. You branch off 8 nodes from each of the 5040 nodes making 40320 nodes for the 8th level and 46233 nodes in total. As you can see each level contains the factorial n where n is the number of that level for example level 2 has 2! nodes in it so the 9th level will have 9! nodes or 362880 nodes and the tree will have 409113 nodes in total. The 10th level will have 10! nodes in it the 11th will have 11! nodes in it and so and so forth by the time we get to the 100th level there will be 100! nodes in that level or 9.33262154537 × 10^157 nodes. Too find how many nodes are in the tree up you just take the sum of the factorials up to a certain value n. For example lets say you want to find how many nodes are in the tree up to the 40th level? To work this out you just take the sum of the factorials from 1! to 40! so 1!+2!+3!+4!+5!+6!+7!..39!+40! Too work out how many nodes are in the tree after that Googolplex level you just take the sum of all the factorials from 1 to 10^(10^100) so 1!+2!+3!...+(Googolplex-1)!+Googolplex! I have decided to call this tree the factorial summation tree as it is based on the summation of factorials up to a certain value x. The function for this is Fst(x)the Fst stands for factorial summation tree. What the function does is processes a certain value x say 5 and gives you out how many nodes are in tree by that particular so in the case of 5 it is 153 so Fst(5)=153.