It would be very difficult to harm a "great big tree" with a weed killer. Anyone that has ever attempted to kill a stump will attest to this. My advice is to either pull out the plants or spray the plants and only the plants.

PS. add a tiny bit of dishwashing soap to your weed killer mixture to help overcome waters surface tension and shake. It will latch on better to the leaves. If the plant is growing up the tree. Cut a segment out to kill the upper portion. Use the search with the Q&A board not the Search listed under Control above. It will help identify additional variations of this answer which may be more specific.