Thread: 2-variable Generating Function

I'm having trouble understanding a function, in relation to binary rooted trees
Def'n: a binary rooted tree is a tree with a root node in which every node has at most two children.
Def'n: A terminal is a node with no children
Let T be a tree, B the set of all BRTs, n(T) counts the number of nodes in T, z(T) counts the number of terminals

Let

Noting the recursive structure of BRTs, let L be the left subtree, R the right subtree

if either L or R not empty, 1 if both are empty


Thus (this step I don't understand)




I'm having trouble understanding how the function is broken down.
Help please?

Hey I-Think.

Are the sets L and R disjoint?

L is the left subtree of the BRT T
and R is the right subtree of the BRT T.
So yes they are disjoint

I should have asked this before, what exactly is T(x,y) measuring? I understand what the other functions are trying to do but I don't understand what T(x,y) is: is this just a vector to calculate both the number of nodes and the number of terminal nodes?

Yes, for a given binary rooted tree T, T(x,y) records the number of nodes in the tree (via x variable) and the number of terminals in the tree (via the y variable)

In terms of your notation, this is really confusing since you it is recursively defined by its written like a one-dimensional algebraic representation that is painful to follow.

If you could re-write this it would be much appreciated, but can you explain what x and y is vs x^a and y^b? Again the way you have written this is completely confusing as hell and doesn't make much sense.