A few yards away the same type of plant grows in bush form, much denser and leafier, but with no berries though both plants bloomed together in May. The little tree has also sent out a good many shoots from the ground, making it clear that the single-trunked form is not necessarily its natural habit of growth.
In fact, I had been entirely on the wrong track, looking in the Old World for a tree from the New. As far as I can tell, the two plants, tree and shrub, are specimens of Schinus terebinthifolius, the Brazilian Pepper Tree. The small white flowers, the growth pattern of the little pepper-type berries (quite different from the Pistacias), the shrubby tree form, and the dioecious nature of the plants all appear to be correct for the Schinus. I can say at least that the leaves are sufficiently like Pistacia terebinthus to earn this Pepper Tree the species name of terebinthifolius. But I can't say much more, except that it took the picture database available on the web to work this one out!
Here are a few more pictures. I enjoy photographing the berries though with their soft colouring they took a little extra camera practice. At least there has been plenty of practice time; the tree holds the berries for many months. I have yet to catch a good picture of the tiny flowers. As you can see, the new growth is a good bronzy green colour.
Definitely not fruit from the Pistacia family! These trees are, curiously enough, from the cashew family, Anacardiaciae.