Answer: To some extent the answer to this depends on what kind of trees they are and how shady it is under them as well as how much root competition there is where you are trying to plant. Honeysuckle might survive there but I would not expect it to bloom prolifically in a shady location. It will also stay thin and sparse due to lack of light. It might however climb quickly as it searches for more light and stretches toward it. Generally speaking, most vines prefer full sun or at least a half day of direct sun.
I wish I could be more encouraging, but it is often difficult to plant successfully under established trees. English ivy will grow in a shaded and difficult spot and will cover a trellis or fence; if you water it carefully the first year or two it will also compete with tree roots. Virginia creeper would also probably manage to grow fairly well there, it might even climb the trees as well as trellis and create a lovely splash of red fall color.
There is a climbing hydrangea that grows well in dappled light under shade and in evenly moist yet well drained soil, however it will not do well competing against tree roots so I wouldn't recommend it here.
There is a bamboo that will tolerate shade, however it is a slow grower and forms a limited clump so it will not give you the screening you are hoping for. Most bamboos need full sun, they are also slow to establish before they put on the growth they are so famous for. (I would not recommend planting a running bamboo in any residential space for that reason.)
In my experience it is usually not very satisfying to see a leggy vine scrambling along trellis. Instead, you might want to invest in an attractive fence and soften it with the ivy or Virginia creeper.
Another option, rather than try to fence the property line, is to plant an island of shrubs closer to your deck or patio or window that you want to protect. If you have space, this might allow you to plant further away from the tree roots in a location more amenable to a wider variety of plants. Another option is to put an arbor or pergola structure over the sitting area and allow a vine to drape over it, thus giving the illusion of shelter.
I should mention that when you are trying to stop noise, the best barrier is an earth berm with evergreens growing on it; unfortunately you can't install a berm over existing tree roots without risking killing the trees. Sometimes you can distract yourself from the traffic noises by using a small burbling fountain at the seating area. This helps you focus on the close-up rather than the exterior noises.
I hope this helps.
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