Both of these species are excellent in permaculture settings, especially as biomass accumulators. They will grow back vigorously after being cut back. We could use the "chop and drop" method, just lopping off the top of the plants and dropping the branches on the ground to leave as a natural ground cover and edge for native species, or we could keep the wood to use for firewood, trellises, mulch, or hugelkultur.
For some reason I have always really liked the trees of heaven. They are quite beautiful and have a tropical-type appearance, and since we really don't have any other trees on the property yet, we have been very hesitant to cut them down. We also learned that the roots, leaves, and bark of the tree of heaven have been used for thousands of years in Traditional Chinese Medicine. Even modern pharmaceutical companies are researching the trunk and root of this tree as a potential drug. We may experiment making a simple tincture using the root-bark. Outside of gathering a bit of root bark and lopping off the top of the trees, we are allowing the trees of heaven to stay put.
At one time, autumn olives were widely planted by the U.S. Soil Conservation Service for purposes of attracting wildlife and serving as a windbreak. Plants can start bearing fruit in their third year, and just one mature plant may produce several pounds of edible fruit. The ripe, red fruits, which are extremely high in lycopene (15 times higher than tomatoes) are especially favored by birds, who readily scatter seeds. Some farms are even planting autumn olive as a cash crop.
Need more reasons not to hate autumn olives? They are also great nitrogen fixers and nectar producers. We are considering including autumn olives in the fruit tree guilds of our soon-to-be planted food forest. Get more info on the value of autumn olives as a cash crop here.
Since both of these plants are considered to be invasive and threatening to native species, we would not likely order in autumn olives or trees of heaven and introduce them to our property. But fortunately for us, they are already here.
If you can't beat them, join them!