That depends a lot what sort of "ordinary people" you're talking about.

Sure, the average appartment-dwelling modern cityfolk may not know much about trees, unless it's a special interest. Your average suburb-dwelling and contryside folk already know more: he/she grows trees in the backyard.

Your average countryside or forest-roaming pre-modern folk, ie: your average Fantasy series character (with exceptions), however had to know a lot about trees. It's very simple: they used so many of them. The bark of this one is good for this, the sap of another for something else, the wood of another is good to make this and that but not this or that. This tree's wood burns fast and hot, but that one is better for a slow burning. That tree grows very fast, and that other one (like oak) takes forever. That kind of tree means this sort of bird (eg: pheasants or partridges) is more likely to be abundant around, and observing trees through the seasons tells you a lot about the weather that year. A lot of this used to be common knowledge especially outside cities. It's a trait of modern men and women that we've lost a lot of traditional knowledge as we get more and more remote from the sources. We no longer know as much about flora and fauna, since we rely on people who do to take care of feeding us and making the objects we use. Most of us would be completely helpless if we had to make tools and objects, grow or hunt our own food etc. Most of us would have not the strictest idea how to balance a garden to get food through all the summer, which kind keeps well which sort of tomatoes makes the better preserves or can ripen on the shelf and which will spoil very fast, what kind of potatoes last longest and so on. Even our grandparents still knew much of this stuff, it's all vanished with refrigerators.

If you look at WOT for example, Jordan kept most of the very specific observations about trees largely to specific characters. Nynaeve and Perrin in particular know a lot about trees and notice them a lot in their POVs, though none of the TR folk is hapless in this area: they are countryfolk. Elayne and other city-dwellers like Moiraine barely ever mentions them. There are also many examples in the series where the characters can't recognize some species.

As for their prevalence in Fantasy it owes much to their prevalence worldwide in myth and religion. They're majestic, long-lived, strong, they have a very clear seasonal cycle etc. Trees and the observation of trees have long been the source of excellent metaphors. Obviously their importance has lessened some in modern times, like much that used to be part of daily life that isn't any more. Trees still make a strong impression on a lot of people, though - especially those who have the opportunity to observe them a lot. Speaking for myself, trees are one of those things I'm really not surprised to find so present in myth and legends.