Total number of nodes in a tree data structure?

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.

Ok, each node has about N subnodes and the tree is L levels deep.

With 1 level, the tree has 1 node.
With 2 levels, the tree has 1 + N nodes.
With 3 levels, the tree has 1 + N + N^2 nodes.
With L levels, the tree has 1 + N + N^2 + ... + N^(L-1) nodes.

The total number of nodes is (N^L-1) / (N-1).

Ok, just a small example why, it is exponential:

                    [NODE]
                      |
                     /|\
                    / | \
                   /  |  \
                  /   |   \
            [NODE]  [NODE] [NODE]
              |
             /|\
            / | \