The shoe horn stops you from squelching the heel counter of your shoe when you put it on. It helps if you loosen the laces thoroughly (especially on boots) as you are then not fighting with your laces to squish your foot into the shoe … not good for foot, shoe or back muscles. If you want a portable shoe horn, buy a cheap plastic one and cut it in two. You really only need the business end of it and half a shoe horn fits in the trouser pocket rather well, without causing an ostentatious bulge!
Shoe trees should be in your shoes whenever your feet are not. They stop the natural wrinkles that occur when you walk (especially on the vamp) from being “set” into your shoes as they dry and cool. That’s why it’s vital to get the trees in as soon as you take your shoes off. Buy decent wooden ones for normal use and cheap plastic ones (Ikea?) for travelling.
By the way – this is what gives rise to the advice that shoes should not, if possible, be worn two days in succession, to give them time to dry out and the shoe trees to do their job. Shoemakers would say this, as it means you need more shoes for the same number of feet(!) but it really does work.