  Given a permutation $\sigma$, its corresponding binary search tree is
obtained by recursively inserting the values $\sigma(1),\ldots,\sigma(n)$ into
a binary tree so that the label of each node is larger than the labels of its
left subtree and smaller than the labels of its right subtree. In 1986, Devroye
proved that the height of such trees when $\sigma$ is a random uniform
permutation is of order $(c^*+o_\mathbb{P}(1))\log n$ as $n$ tends to infinity,
where $c^*$ is the only solution to $c\log(2e/c)=1$ with $c\geq2$. In this
paper, we study the height of binary search trees drawn from the record-biased
model of permutations, introduced by Auger, Bouvel, Nicaud, and Pivoteau in
2016. The record-biased distribution is the probability measure on the set of
permutations whose weight is proportional to
$\theta^{\mathrm{record}(\sigma)}$, where
$\mathrm{record}(\sigma)=|\{i\in[n]:\forall j<i,\sigma(i)>\sigma(j)\}|$. We
show that the height of a binary search tree built from a record-biased
permutation of size $n$ with parameter $\theta$ is of order
$(1+o_\mathbb{P}(1))\max\{c^*\log n,\,\theta\log(1+n/\theta)\}$, hence giving a
full characterization of the first order asymptotic behaviour of the height of
such trees.
