Pretty familiar with Monticello. Grew up 30 miles from there. James Monroe's place, Ash Lawn, in on the next ridge to the south.
All of the founding fathers' estates in Va. have something to offer botanically, since most were working plantations in their time. Jefferson's is among the best for architecture and botanical history. Mt. Vernon has some original trees planted by George W. --two immense tulip poplar are the most visible. The boxwoods I post here occasionally are on the grounds of George Mason's Gunston Hall plantation just outside the city. James Madison's Montpelier plantation, further downstate, has one of the best old growth tulip poplar in the U.S. and is located one of most beautiful sites of all the old plantations in Va.
There are hundreds of notable trees around the Capital area that have less famous pedigrees though. The most famous are the cherry trees, but there are others. The two southern magnolias snug up against the White House on the south lawn were planted by Andrew Jackson, for instance. The Capitol building's grounds has dozens of "historic" trees.
Washington has been known as the "City of Trees" because there are so many big old trees around. a fact we're reminded of every winter and summer when ice and thunderstorms topple 150 ft. oaks, elms and sycamores onto houses.