For grown-ups, a demand for actual waterproof shoes has driven Kamik to create their Atrium Collection. Imagine rubber boots or fly-fishing waders distilled 
I have taken them on rigorous trail hikes, run with them on track and on pavement, and trooped all over NYC and beyond, never with any discomfort. The shoes 
Hiking boots also have a more rigid upper and are almost always a high top cut. They may also come with a waterproof breathable layer such as Gore-tex, 
Waterproof mascara. Pantyhose. Inappropriately high heels. Hours passed. My feet hurt. Really, really hurt. On the lurch home, I wasn't so much dragging my 
Which brand of hiking/trekking shoes is cheapest?
A good pair of boots is not something to scrimp on. A good pair of boots may cost twice as much or more than other "cheap" boots, but they will provide much more comfort and be easier wearing on your feet. Many inexpensive boots can cause pain and discomfort, plus, they will probably not last as long as high quality boots. I personally tried cheap boots once a few years ago, and I wound up killing them in one trip. The waterproofing leaked and the glues in the boot melted. They only lasted for 60 miles and two weeks, so I was rather disappointed.
Cheap and waterproof do not really go well together, and while there are inexpensive waterproof boots, the waterproofing is going to be questionable and not to be relied upon.
REI is a great place to test what boots fit well, and the staff is well trained and knowledgeable on the products they sell. They can also help you find the right type of boot for the terrain you'll be in primarily. Garmont makes boots that I like and have found to be long lasting.
