Of those many books that I have read about the art of growing bonsai trees, only one goes sufficiently in-depth about this issue. Unfortunately, that one book is probably only available in Czech. I will not go too much into depth here, just the basics for your own research should you wish it.
The distinction between a tree and a bush/shrub is not clear-cut, but there are properties that some plants have that make them definitively shrubs – like roses – or definitively trees – like spruces. A rose, no matter how big and old will be unruly and bushy. Spruce, no matter how small, will be a tree, with a definitive main stem.
The most important factor in this is the absence or presence (and strength) of so-called apical dominance. In plants with strong apical dominance, the main stem produces hormones that regulate and/or inhibit the growth of secondary branches. That is the reason why spruces almost invariably have one upright stem with comparatively thin branches – they have very strong apical dominance like most conifers do. Roses, on the other hand, do not have apical dominance worth speaking of. They continuously sprout new twigs from the base of the stem near the roots. Common hazel is somewhere in between – at the start it has strong apical dominance, but the bigger the main stem gets the smaller inhibiting influence it has, it slows its growth and inevitably suckers start to sprout from the base even when the main stem is perfectly healthy and strong.
Apical dominance applies to roots too. Some plants tend to grow long, thing, non-branching roots, some have bushy ones.
It also changes during each season – in many plants with leaves the hormones are produced by the leaves, thus the more leaves there are on the branch/stem, the more it inhibits those below it. That can be used to your advantage for certain species – more on that when talking about them.
For a beginner, plants with strong apical dominance should be avoided, especially fast-growing ones. Many such plants when cut do not branch out but simply the bud nearest to the cut continues as the main stem. Some might even die. Best are plants with some apical dominance, but not a very strong one. But also not so weak as to tend to sprout suckers each season.
That, unfortunately, means that for a beginner or a small-scale grower, the most iconic of all bonsai trees – pines – are not suitable, as well as most of the coniferous trees overall. They are the most difficult to manage and to form, and some of them are downright snowflakes. Best suited are leafy plants, with moderate growth rate.
I will go into more detail when talking about specific species in due course.