Trees have both perennial roots that under normal circumstances don't die. They also have an annual root system that dies every year and regrows. The annual root system can extend out many times farther than the crown of the tree.
Here is a good article from NCSU that explains tree anatomy.
I can't say that I agree with the statement that trees have annual roots. Trees have two main classification of their roots: Their main large roots (lateral, tap, striker or heart roots) provide support for the tree, transport mechanisms for water/nutrients/photosynthate etc, and storage of the same things they transport. These large roots are perennial, woody, and they do extend out well past the dripline of the tree and usually stick around until some form of major damage occurs. Trees also have absorbing roots, which are responsible for absorbing water and nutrients from the soil, excreting exudates (carbs made by the tree) into the soil, and root tip extension. These absorbing roots are ephemeral, and will only grow into areas where conditions are correct for absorbtion. These roots will die quickly when conditions become unfavorable (too dry, wet, cold, hot, salty, etc) and contribute large amounts of organic matter to the soil. If conditions are favorable for a long enough time, some of these absorbing roots will become part of the expanding lateral root network.
Most of this information is a quick paraphrase of a general overview of tree root systems beginning on page 86 of Up By Roots by James Urban
With all that in mind, you could definitely argue that trees do "shed" their absorbing roots locally whenever conditions don't support those roots. It has also been shown that trees will lose roots in response to drastic pruning or injury. The loss of root mass due to injury is also a localized response related to where the injury occurred.
"Instead of Pay It Forward I prefer Plant It Forward" ~Howard Story / "God has cared for these trees, saved them from drought, disease, avalanches, and a thousand tempests and floods. But he cannot save them from fools." ~John Muir