Do height-balanced binary trees have logarithmic depth?
Let us denote by $N_h$ the minimum number of leaves of a tree in your family of height $h$. Clearly $N_0 = 1$, and
$$ N_h = N_{h-1} + \min_{0 \leq d \leq c} N_{h-1-d}. $$
(Here $N_h = 0$ if $h &lt; 0$.) You can prove inductively that $N_h$ is monotone, and so for $h \geq c+1$,
$$ N_h = N_{h-1} + N_{h-1-c}. $$
This is a recurrence relation whose solution is $N_h = \Theta(\alpha^h)$, where $\alpha$ is the maximal real root of $x^{c+1} - x^c - 1= 0$; for example, when $c = 1$, $x$ is the golden ratio.

Since the number of leaves in a tree of height $h$ is $\Omega(\alpha^h)$, it follows that a tree containing $n$ leaves has height $O(\log n)$.