Efficient random sampling of binary and unary-binary trees via holonomic equations

We present a new uniform random sampler for binary trees with $n$ internal nodes consuming $2n+\mathrm{\Theta }(\log (n{)}^2)$ 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 $\mathrm{\Theta }(n)$ random bits on average. Both are the first linear-time algorithms to be optimal up to a constant.