The city of Paris has an online interactive map of the "remarkable trees" of Paris, including the Bois de Vincennes and the Bois de Boulogne. The locust tree in the square Viviani, just across from Notre-Dame, is considered the oldest tree in Paris. It was planted in 1601. Here's the tree in 1854 and today.
According to the interesting Histoires de France racontées par les arbres*, this "oldest tree in Paris" is... American! A Virginian, to be precise.
My favorite tree in Paris is in the lovely Pré Catalan, in the Bois de Boulogne. I was there a few days ago in the middle of the day. It is home to the Theater in the Shakespeare Garden and to a very chic restaurant. The roads around it, even at mid-day, are also lined with prostitutes lurking behind trees, and their shifty-looking clients. Luckily the park itself was so empty you could hear only the wind rustling in the leaves and the wheels of two children on their scooters.
My favorite tree at the Pré Catalan has always been the enormous, six-meters-around silver-barked beech tree at its center. It is at least two hundred years old. But the beech tree had suffered since the last time I saw it. A huge gash in its side shows where a heavy branch has fallen.
The tree is now surrounded by a fence, and you can't go near it. A sign attached to it gives you the sad news: "Safety fence. The beech is being attacked by an irreversible wood rot caused by wood-eating fungus. Risk of falling branches." Sic transit gloria arbori.
* The history of France, as told by trees