I'd love to compile a list of vegetables, or vegetable like fruits which grow on trees. I'm particularly interested in those that grow in a temperate climate. Perhaps we can note those with a (T).

Avacados
Unripe Jackfruit
Linden Trees - the leaves (T)
Moringa oleifera - leaves and young seed pods

and I'm stuck.

Mulberry leaves.

You don't have to be a silkworm to eat them, they make good animal fodder and are similar to grape leaves in texture and cooking applications. With the help of Google, I was even able to find a recipe for Stuffed Mulberry Leaves with Chicken.

Heart of palm, AKA "swamp cabbage"

various edible bamboo.. stretching the "tree" aspect slightly as it is the new shoots that you eat.

Some of the nut trees make great staples for carbohydrates and protein. Sweet chestnuts have been cultivated on a massive scale in Corsica since roman times and the nuts collected, dried, and stored for making all sorts (up to bread products).

A good book to look at for inspiration is "Tree Crops". Very old, but you can find pdf versions of it online.

Young Bech leaves are lovely

Thanks for the replies!!

I'm excluding nuts and seeds from the veggie category. I do love chestnuts though. I planted 125 this year.

Camillia sinensis Tea plant.

I've heard of people pickling the fruits of the tulip popular but have never done it myself.
You can drink the nectar from the tulip tree also. The same with the honey suckle.

Sweet birch bark can be harvested to make a wintergreen flavoring and a beer.

sourwood leaves are edible. But not particularly tasty.

Polk salad is pretty much a tree down here in the south... The new leaves shorter than your finger can be boiled in a few changes of water and taste just like cooked spinach.

Most nut tree mast I myself would consider to be a vegetable. It can be toasted and ground up and made into a flour or boiled into a mush.

Shagbark hickory bark can be boiled into a syrup.

This is an important, and fairly short, list.
Rose-of-Sharon young leaves and tips are good. I have read that any Hibiscus is in fact edible, but that deserves some corroboration.
Multiple sources say that cassava leaves are edible cooked.....and this does reach small-tree size. Plants like this (including chaya and taro) are important in the tropics because they are toxic raw---and therefore ubiguitous livestock leave them alone, while still providing nutritious greens to people when cooked.
Flowers of several things, including black locust, wisteria, and the redbuds, are edible (the first two when cooked, and redbud either raw or cooked).

Does Hibiscus tolerante frost?

Rose of Sharon handles zone 7-8 with no problem.
It's a pretty popular landscaping tree down here.

I live in zone six,rose of Sharon grows like a weed,the flowers are edible, bland,and usually occupied by a bee,so be careful when you pick them. The prairie mimosa has edible seeds, my mimosa tree makes lots of seeds but is not documented as being edible except for its flowers,which are edible but not tasty.

Yellowhorn has edible leaves, flowers, and nuts. Honey Locust produces high protein pods that might be eaten as a green vegetable when young. Black Locust also produces flowers that are edible, and the foliage can be used as animal fodder. I've read that sweet potato leaves make a good green vegetable, but haven't tried them yet.

One wild plant in my location is Miner's lettuce, which is dead-simple to identify, even by little children. I like to stir-fry it like spinach with anchovies and a little oyster sauce. Milkweed and nettle are also present in my area, but I havent't gotten around to trying those yet. Miner's lettuce is very, very high on my importance list, because I start seeing it in January near the base of the mountain on the southern sides, where as I can find it in June on the north slopes at higher elevation. That's the one green vegetable that is easy to find during the period of the year with the least number of fresh choices.

Rose of Sharon is at least zone 5.  Weedy here.
        In the humid tropics, Ensette or Abyssinian banana is a staple starch.  The leaves can be used for wrapping like true bananas (Musa), but it is the pseudostem and corm that provide the calories.  Not technically a tree, but does get that high.
       Unripe papaya (another pseudotree).
       Toona/Cedrela sinensis (vegetable mahogany) for temperate areas to zone 5.  Young leaves are cooked, after being stripped from their midrib, bruised and salted, especially in pajeon (Korean "scallion pancakes," but with Toona rather than scallions).
      Baobab: leaves are edible (so is the fruit).  Portulacaria afra ("elephant food," Afrikaans name seems to translate as "lardbush"): leaves are edible, best in the afternoon (too tart in the morning).  The young cladodes/pads of all prickly pears (Opuntia section platyopuntia & Nopalea) are good cooked or pickled vegetables once you get the thorns & glochids off (flowers and fruit also edible, same warning); many of the tropical ones are aborescent.  These are tropical arid zone trees.
        Pereskia aculeata (great fruit, but the leaves are edible too) is really a rampant, thorny liana, but other Pereskia with more rigid stems can probably be used the same (fruit less famous, or maybe the plants simply less invasive).  Thornscrub (semiarid or seasonally dry tropics).

The trouble with eating trees is reaching the harvest.  In Ensette, the whole stem is taken down after about 4 years, then pounded or somesuch to extract the starch.  Hibiscus can be grown as a shrub.  Portulacaria is small as a houseplant.  But with true trees (other than sap crops like maple & birch), you pretty much have to espalier or coppice/pollard them to eat much.

Chaya

toona sinensis

I believe both the leafs and flowers of the linden tree are edible.