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

If your tree is approximately full, that is every level has its full complement of children except for the last two, then you have between N^(L-2) and N^(L-1) leaf nodes and between N^(L-1) and N^L nodes total.

If your tree is not full, then knowing the number of leaf nodes doesn't help as a totally unbalanced tree will have one leaf node but arbitrarily many parents.

I wonder how precise your statement 'each node has about N nodes' is - if you know the average branching factor, perhaps you can compute the expected size of the tree.

If you are able to find the ratio of leaves to internal nodes, and you know the average number of children, you can approximate this as (n*ratio)^N = n. This won't give you your answer, but I wonder if someone with better maths than me can figure out a way to interpose L into this equation and give you something soluble.

Still, if you want to know precisely, you must iterate over the structure of the tree and count nodes as you go.