This is a cinnamon tree (Cinnamomum zeylanicum). We use the bark as a spice in North America, but they also use the leaves in the Caribbean both to make tea and to flavor stews like you would with a bay leaf (Laurus nobilis).
Picking a leaf off the tree and crumbling it in your hand releases the most wonderfully fragrant cinnamon smell. I came to love the leaves above all other parts of the plant and beyond most other spices too.
The deep ridges down the length of the leaf made the tree very easy to identify, unlike West Indian bay (Pimenta racemosa). I must have passed thousands of these trees in my travels but was almost never able to distinguish it from several other tropical trees with similar shiny foliage. For that reason I rarely picked bay leaves, while I regularly stuck a cinnamon leaf into my pocket whenever I saw one.
I haven’t been back a week and I already miss it terribly.