Let $T$ be a complete binary tree with $t$ terminal vertices (or leaves). I want to know the maximum height $h_\max$ it is possible for $T$ to have. The only result I know is that if $T$ is a binary with heigh $h$ wth $t$ terminal vertices, then $$\lg t\leq h.$$ But this isn't useful since we have a lower bound for $h$. However, my reasoning is the following. The way to make in a tree more height with the minimum quantity of terminal vertices is this way:
$\hspace{3cm}$$\hspace{2cm}$
and so on...Hence, if we have $t$ terminal vertices, the highest height possible is $t-1$. This is quite a counting argument and not sure if too formal, is $h_\max=t-1$ right?