Binary search trees (BST) are a popular type of data structure when dealing with ordered data. Indeed, they enable one to access and modify data efficiently, with their height corresponding to the worst retrieval time. From a probabilistic point of view, binary search trees associated with data arriving in a uniform random order are well understood, but less is known when the input is a non-uniform random permutation. We consider here the case where the input comes from i.i.d. random points in the plane with law $\mu$, a model which we refer to as a permuton sample. Our results show that the asymptotic proportion of nodes in each subtree depends on the behavior of the measure $\mu$ at its left boundary, while the height of the BST has a universal asymptotic behavior for a large family of measures $\mu$. Our approach involves a mix of combinatorial and probabilistic tools, namely combinatorial properties of binary search trees, coupling arguments, and deviation estimates.