Most every street and alley in St. Louis has one or more (or all) of these trees. There are other trees around with similar leaves, but these four were difficult for me to distinguish once upon a time. Identifications can be fairly easy if a tree is in flower or fruiting. Many people can instantly identify a walnut nut or a sumac berry cluster, but fruits and nuts are usually on the trees for a limited time. For most of the year, we have only leaves and bark to help us make identifications. Bark can be really tricky (it all looks alike at first), so I'm discussing only the leaves here.
This picture shows one leaf each from (left to right): tree of heaven, smooth sumac, golden rain tree, and black walnut.
One leaf each? Yes, these leaves are pinnately compound, meaning each leaf is made of a central stalk with a row of small leaflets attached to each side. This can be confusing, but leaves sprout out of the buds that are arranged along a tree's twigs or its stem. The growth that comes out of the bud is the leaf, whether it is a simple leaf like an oak or a maple, or whether it is a compound leaf like these. Another way to say it is that the bud is located at the base of the leaf, and for these compound leaves, the bud is located at the base of that central stalk, and not at the base of each leaflet. However, I'm not a botanical authority so somebody might disagree with my explanation. Anyway, after a while you get used to distinguishing between them and you don't worry about it that much any more!
Many sources will tell you how many leaflets are found on each leaf of these four species and how large each leaf should be. Guess what? There is not one number or one size for each tree. There are ranges, and the ranges overlap -- perhaps not the most helpful markers for making a casual ID. Some sources will tell you that the leaf of one species always ends in a pair of leaflets while some other species ends in a single leaflet. I don't think these trees read botany books -- or maybe they just don't care to obey botanists. In any case, you might find a single or double leaflet at the tip of any of these, as I have.
Looking at the leaflet margin does help. The leaflets of tree of heaven (below, left) have a mostly smooth margin, with one or two larger bumps at the base. Walnut leaflets (below, right) are also smooth, but do not have those bumps, and the curve continues smoothly, uninterrupted right to the base of the leaflet.
Both of these leaflets can feel slightly fuzzy, but I think the walnut feels fuzzier sometimes. In the right light you can sometimes see tiny sparkles on the topsides of walnut leaves. Something very small there reflects the light (seen as tiny white dots on the leaflet below). I also find that the little tips of walnut leaflets are longer than on the other species here. The tips are drawn out to a longer, narrower point.
The two that were most difficult for me to learn to distinguish were tree of heaven and smooth sumac. The stems and trunks look a bit similar, and young tree of heaven individuals can be similar in size and shape of canopy compared with smooth sumac. I've noticed in St. Louis that they sometimes grow up together in the same places. As noted above, the serrated margin of the smooth sumac sets it apart. It may be helpful to look at the leaves from their reverse side, especially if you think you have both species present and can't decide which is which. I'm always surprised at how light the underside of the smooth sumac is, almost white sometimes.
Below, left to right: underside of tree of heaven, smooth sumac, golden rain tree, black walnut
Shortcut: although fruits and nuts are not on the trees for long, look around in the leaf litter (or amongst the fast food wrappers and beer cans) at the base of the tree. Walnuts turn black and stay on the ground for a year or more. Golden rain trees have little papery brown capsules that last for months. The reddish berry clusters of smooth sumac harden and turn brown, and some will fall to the ground. If you find any of these you know the parent tree is probably just over your head. Tree of heaven produces little wind-blown paper samaras, so if you're poking around under those trees for off-season clues you're out of luck.