What are the vegetables that grow by the tree?

Leaves, bark, roots and branches are vegetative aspects on trees.

Mushrooms?

Lime leaves are common ingredients in Thai recipes.

If you’re asking about vegetables that grow in the form of trees, I don’t know of any.

The only food items that I know of that we get from trees are fruits, not vegetables. There are “pepper trees”, but I don’t think that the pepper pods that grow from them are food items; at least they aren’t where I come from.

@CWOTUS Pepper pods are the fruit of the pepper tree.

Bay leaves are aromatic tree leaves and used in cooking. Two of these bay leaf tree species are Umbellularia californica and Laurus nobilis.