This question seems worthy of debate in a forum for herb debate. 

Not that I always give undying credence to sources like a Webster's on-line, but at least as an example of a formal opinion to help get the debate ball rolling, here is what they/it says.

1: a seed-producing annual, biennial, or perennial that does not develop persistent woody tissue but dies down at the end of a growing season

2: a plant or plant part valued for its medicinal, savory, or aromatic qualities

By definition, Webster appears to exclude trees from herbs, at least until one looks up their definition for "plant" [noun]

1 a: a young tree, vine, shrub, or herb planted or suitable for planting b: any of a kingdom (Plantae) of multicellular eukaryotic mostly photosynthetic organisms typically lacking locomotive movement or obvious nervous or sensory organs and possessing cellulose cell walls.

Prior to 4 years ago, give or take, I generally would have characterized herbs as a variety of dried plant material used to flavor foods. That was the period before I developed an interest in learning the benefits of herbs as natural medicines and or foods. My perspective has changed the past several years. Retroactively, I would also characterize my past understanding of herbs as generally one and the same with "weeds". One thing I've come to learn the past few years is that I once referred to patches of plants broadly as "weeds" because that was how I was taught. How many other people were taught to refer to plants as "weeds" mainly because they otherwise did not know how to identify them one from the next?

For years I have enjoyed what is commonly referred to as "mowing the grass", but in recent years have come to discover that my lawn front & back is comprised as much by various herbs - some I can now easily identify like Dandelion, Plantain and Clover, and several I still have not identified - as it is actual grass. These days, my neighbors might think I'm a bit cuckoo for the way (pattern) I mow the lawn from one week to the next. Especially in the early to late spring when there are all kinds low-growing flowering things, I will leave a given clump of growing stuff uncut, even when it is a patch out in the middle of an otherwise freshly mowed yard. I'll leave it uncut for a while, partly just to see where it goes, but also because some of the other neighbors - like bunnies, birds and other undomesticated critters, like to snack on AND OR lounge in these clumps of various weeds. Right now (late spring), there is a broad patch of bright yellow "butter cups" growing near the far end of the back yard, and an equally broad patch of uncut blue flowery thingies growing under some tall pine trees at the side of the yard, and if one looks closely down into the grass in patches here and there - even after the grass has been mowed, one can see tiny wild strawberries that have been ripening the past week or two. Sometimes I'll even leave a small clump of grass uncut. Eventually it gets quite tall and goes to seed. Is grass an herb? (Kentucky Blue, Fescue, that kind of grass)

In the back yard there is a patch of thicket growing wild. It is the unofficial property line between my yard and the neighbor's to that side. It is also the most obviously uncared for section of property. Somebody, perhaps a previous tenant, had thrown all sorts of refuse into this patch ... bunches of old bottles, pipes, broken glass, cans and such... a rather peculiar time capsule as it were.  Despite this there is quite a variety of weeds growing in there. During the past  3 years I've been able to identify several of the weeds growing in this patch, but there are several I still have not yet figured out. There is a black walnut tree, a bean tree "Catalpa", a patch of Sumac (staghorn), an increasingly unwieldy mass of wild grape that up till recently had been choking out the better part of the entire thicket, a young Mulberry Tree that is just recently bearing ripening fruit, a good patch of Honeysuckle, some ground ivy, a small stand of Sassafras trees with several young shoots, some Lilac, Forsythia, and a bunch of other green things I have not yet identified including some gnarly vines and various low shrubs. Although it was generally against my character to do so, when I first moved in I surgically put down some wicked "designed to kill everything living it touches" spray in & on this thicket in an effort to eradicate a wide patch of poison ivy and oak. I'm not at all fond of poison ivy. Within a month the entire area sprayed had turned black. From the kitchen window view this spot stood out. There was a hole bunch of various bright green things growing at knee level and above, then this vacant black spot down near ground level.  By the time that first summer ended, new green things had begun to grow out of this black spot, and poison ivy was not among them.

In a way, herbs can be likened to the place that holds them - ground, the way water can be likened to the places that holds it - lakes, rivers and oceans, and the way the space above ground holds air. The thing about herbs is that even though a person may not know how to botanically identify each of them individually, they can at least see how they differ in shape, shades of color and size from one plant to the next. If humans could see the various individual batches of chemical compounds making up one patch of air space compared to a patch next to it comprised of different chemicals, would we still generally refer to airspace generically as "air" or would we have different names for different identifiable patches? So, what is an herb? As far a plants that grow out of the ground, are there any that aren't herbs? Burdock is a good example for debate. In the early spring, the stems can be low cooked and eaten as though a food "vegetable". The young leaves can be used as a part of a pain-free bandage applied to burn wounds "first aid". Wildman claims that the mature bur can be eaten "nut/seed". The roots can be chopped and dried for use in teas and tinctures "medicine". Is it an herb, food, fruit, vegetable, mineral, all of the above or simply a matter of perspective?