Quora 7/15/2015: Based on this definition, a tree is not a fruit. However, many produce fruits.Vegetables are considered any part of the plant that is not a fruit. ... Most trees(oaks, pines, etc) do not have particularly edible stems/trunks, leaves, or roots.

Continuing the thought, eggplant, cucumbers, peppers, avocado, figs, and tomatoes, for starters, are fruit. That part of a plant that contains nutritious pulp and the seed is fruit. (Nuts do not have pulp.)

Vegetables are lettuce, celery, spinach, rhubarb, and all herbs etc.

To answer your question:

Adding to the list already provided here: “Hearts of Palm”, the stem of the palmetto tree, is a vegetable that “grows” on trees, rather it IS the tree and is not picked from the tree.

Mushrooms/Lichen grow on trees, and logs but they are not produced by the tree.

At the moment I can’t think of other examples.
Interesting question though.