  We present a new uniform random sampler for binary trees with $n$ internal
nodes consuming $2n + \Theta(\log(n)^2)$ random bits on average. This makes it
quasi-optimal and out-performs the classical Remy algorithm. We also present a
sampler for unary-binary trees with $n$ nodes taking $\Theta(n)$ random bits on
average. Both are the first linear-time algorithms to be optimal up to a
constant.
