What an odd name. It sure sounds interesting, like some sort of exotic fruit tree. Alas; argyle apple is a eucalyptus; to be specific, Eucalyptus cinerea. Compared to most other eucalypti, it stays rather low. It barely gets as tall as a two story house, even if it gets broad enough to shade most of the backyard. The rusty brown bark becomes roughly furrowed. The irregular branch structure can be quite sculptural.
The main attraction of argyle apple is the aromatic silvery foliage. Young trees are outfitted with circular juvenile leaves that are attached directly to the stems without petioles (leaf stalks). Lanceolate adult leaves are as silvery as juvenile foliage is. (Juvenile foliage of most other eucalypti is more colorful than adult foliage is.) Aggressive pruning of small trees keeps foliage juvenile for a long time.
Actually, those who know how to work with it might pollard or coppice argyle apple. Pollarding eliminates all foliated stems at the end of winter, but for the rest of the year, allows vigorous arching canes of very silvery juvenile foliage to spread outward from a few stout limbs on top of a trunk. Coppicing allows the same sort of growth from stumps just above grade.
Tony Tomeo 
Aka Silver Dollar eucalyptus.
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Eucalyptus polyanthemos is what we know more commonly as silver dollar gum. The common name is sometimes applied to any unfamiliar eucalyptus with silvery round leaves though, including this one.
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I’ll be darned. This is the eucalyptus I met first, and the one that I see used most in dry arrangements. Every time I’ve seen someone writing about eucalyptus, I’ve thought, “Nope. That’s not it.” But this is it — and I’m glad to finally know something about it.
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They are not easy to identify or distinguish from each other. As you have noticed, there are other specie of Eucalyptus that are grown for cut foliage. This one does not happen to be very common for that use, but it is not unheard of either. It may be more common in your region.
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It’s good to know there are some kinds of eucalyptus that aren’t tall trees.
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There are a few eucalyptus that are quite practical for home gardens. They just have a bad reputation because of the bluegum and redgum that were grown for timber, and then abandoned.
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My first time in California, 1967, the eucalyptus trees seemed iconic. Only later did I learn that they’re originally from Australia.
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The bluegum is so common as a naturalized exotic that some want to classify it as a native. I really do not get that, but I suppose it goes along with so much of our other fake history.
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