Total number of nodes in a tree data structure?

Question

I have a tree data structure that is L levels deep each node has about N nodes. I want to work-out the total number of nodes in the tree. To do this (I think) I

Answer 1

Just to correct a typo in the first answer: the total number of nodes for a tree of depth L is (N^(L+1)-1) / (N-1)... (that is, to the power L+1 rather than just L).

This can be shown as follows. First, take our theorem:

1 + N^1 + N^2 + ... + N^L = (N^(L+1)-1)/(N-1)

Multiply both sides by (N-1):

(N-1)(1 + N^1 + N^2 + ... + N^L) = N^(L+1)-1.

Expand the left side:

N^1 + N^2 + N^3 + ... + N^(L+1) - 1 - N^1 - N^2 - ... - N^L.

All terms N^1 to N^L are cancelled out, which leaves N^(L+1) - 1. This is our right hand side, so the initial equality is true.