It’s a really big number, and that’s all the practical significance of it.
The TREE function is defined as:
> Consider sequences of rooted trees with vertices labelled from a set of n labels. The ith tree in the sequence contains at most i vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(n) is defined to be the longest possible length of such a sequence.
Basically, a fast growing function arising out of a graph theory problem.
There’s nothing particularly special about TREE(3), except that it goes to show that TREE is an incredibly fast growing function, such that it starts of slow and unassuming, with TREE(1) = 1, TREE(2) = 3, but suddenly, at TREE(3), it blows up to an incomprehensibly large number, to which there is no term of comparison you’d be able to grasp.
It’s the TREE function that’s fascinating, because it grows so quickly, growing faster than other well known fast growing functions like the Ackermann function, or the function g used to define Graham’s number (with Graham’s number being defined as g(64)), etc. Of course, in the world of fast growing functions, it remains a baby, since it remains a computable function, and is therefore outgrown by uncomputable functions like busy beaver functions.
If you’re interested in the branch of math that deals with this (to no real practical end), it’s called googology, which is the study of large numbers and fast growing functions.
Basically, you can build trees with different color nodes and try to make as long of a sequence of trees as you can without ever having a tree in the sequence contain a previous tree as a sub-tree. With one color, you can only do 1 and with two colors, the longest sequence you can make is length 3, so Tree(1) = 1 and Tree(2) =3, but with three colors, it becomes some incomprehensibly large number. It has been proven though that you cannot make an infinitely long sequence using a finite number of colors.
As for the mathematical importance, there isn’t any really. There are other sequences that grow incomputably fast like the Busy Beaver numbers which theoretically could be used to solve a number of theorems if we could compute them. The tree sequence is just one that’s fun because it grows explosively fast while not being too tough to understand what it’s counting.
Comments
It’s a really big number, and that’s all the practical significance of it.
The TREE function is defined as:
> Consider sequences of rooted trees with vertices labelled from a set of n labels. The ith tree in the sequence contains at most i vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(n) is defined to be the longest possible length of such a sequence.
Basically, a fast growing function arising out of a graph theory problem.
There’s nothing particularly special about TREE(3), except that it goes to show that TREE is an incredibly fast growing function, such that it starts of slow and unassuming, with TREE(1) = 1, TREE(2) = 3, but suddenly, at TREE(3), it blows up to an incomprehensibly large number, to which there is no term of comparison you’d be able to grasp.
It’s the TREE function that’s fascinating, because it grows so quickly, growing faster than other well known fast growing functions like the Ackermann function, or the function g used to define Graham’s number (with Graham’s number being defined as g(64)), etc. Of course, in the world of fast growing functions, it remains a baby, since it remains a computable function, and is therefore outgrown by uncomputable functions like busy beaver functions.
If you’re interested in the branch of math that deals with this (to no real practical end), it’s called googology, which is the study of large numbers and fast growing functions.
Numberphile explained it better than I could: https://www.youtube.com/watch?v=3P6DWAwwViU
Basically, you can build trees with different color nodes and try to make as long of a sequence of trees as you can without ever having a tree in the sequence contain a previous tree as a sub-tree. With one color, you can only do 1 and with two colors, the longest sequence you can make is length 3, so Tree(1) = 1 and Tree(2) =3, but with three colors, it becomes some incomprehensibly large number. It has been proven though that you cannot make an infinitely long sequence using a finite number of colors.
As for the mathematical importance, there isn’t any really. There are other sequences that grow incomputably fast like the Busy Beaver numbers which theoretically could be used to solve a number of theorems if we could compute them. The tree sequence is just one that’s fun because it grows explosively fast while not being too tough to understand what it’s counting.