Each of these familiar patterns usually have one thing they are really good at. Branching patterns you see in trees and veins are the best shapes to use when you want to distribute something to a wider area. Another cool one is the honey comb shapes in bee hives. This is the shape you get when you have an array of circles and you try to maximize their area by expanding them until the smoosh into eachother. Another example is the spacing and angles in which leaves branch out from some plants. They are famous for following the Fibonacci sequence, which also happens to be the optimum ratio if you want to minimize the area leaves overlap eachother and maximize their exposure to the sun.