the height of a tree given n nodes and a condition
The height of a tree is 1 + the height of its highest subtree. Which way of distributing the nodes to the subtrees will allow you to maximize that parameter?


   Put $\frac 13 n$ nodes into the left tree and $\frac 23 n$ into the right one (or vice versa).


If you apply this recursively, what do you get for the height?


   You get the recurrence $h(1) = 1$, $h(n) = 1 + h(\frac 23 n)$, $n&gt;1$. This solves to
 $h(n) = 1+\log_{\frac 32}n$.