Presumably you would like to "take the gift tree with you" when you move. That's probably not very practical, but you can follow a long tradition of taking PART of the gift tree with you.
What is your timeframe? For large trees, you are generally better off to layer (root a cutting, though layering involves doing that while still attached to the tree) rather than trying to transplant a large tree. That takes some time, so if you are not planning well in advance, it can be hard to complete in a timely manner for moving. If you can come to an arrangement with the new owners (make it part of the sales contract) you might be able to get that started before you sell the house and put it in the contract that you will finish (remove the layered cuttings) sometime after the house sale, if time is short. You will probably need access for you or a designated person to keep the layers watered until they are ready to go.
You can also try rooting regular cuttings, but that's typically pretty hit or miss.