The best or worst piece of advice is "consider it early enough". That is, avoid difficult shapes and fragile parts already when you buy or make your trees (or any other kind of gaming terrain, for that matter). In the long run, preferring robustness over spectacularity will pay off.
If it's too late for that then, well, sturdy boxes, tall enough for vertical storage. For some types you could also try cardboard dividers that can be found in glassware packaging, moving supplies and such. They're not that difficult to make either. Those may help against full scale pile-ups from tilting and bumps.