I'm hoping that someone can shed some light on this running time.
I have a "tree", for lack of a better description, that has a length $l$ and depth $d$. I want to maximize the tree size, which we'll call $N$. The tree size is $$N=l^d$$
I also want to minimize the running time $t$, which is $$t=l \cdot d^2$$
I'd like to determine the fastest running time for a given $N$. That is, I'd like to find a formula for $t$ in terms of $N$. What is this formula?
I thought that we may be able to use calculus to somehow maximize the ratio $N/t$. After that, we can divide by $t$ to get the formula. But I don't know how to do this with an equation like this.
Possibly someone knows of something similar that has already been analyzed.