“I’ve heard that cutting through the roots around the drip line of a tree or shrub with a shovel can encourage it to flower. Is that true?” -Ben
The technique you’re referring to is a rejuvenation practice known as “root pruning,” in which roots are cut in order to control growth. While it can work, it wouldn’t be my first choice. I’d start by making sure the plant is getting adequate light, water, and nutrients before resorting to such extreme measures.
Think of it this way: when you cut the roots of a shrub or tree, the plant draws in upon itself and puts its resources toward its own survival and continuation of the species. It does this by increasing flowering and fruiting and by compacting both its root ball and its growth.
Root pruning is normally used to strengthen and prepare trees and shrubs for the shock of transplanting, and for training container plants (well-timed root pruning is an integral part of the practice of bonsai).
If a tree or shrub is overgrown and no longer blooming, root pruning may indeed rejuvenate it. Using a sharp shovel, make deep cuts into the soil around the plant in a circle at least 2-3 feet in diameter. Some gardeners cut the entire circle; others make cuts with spaces in between.
Remember that the more roots you cut, the more stressed your plant will be, so compensate with regular watering for several weeks until new roots form. Prune roots in late fall or early spring when the plant is dormant.