[Edit: Two quick downvotes let me fear that my answer is likely to be misunderstood. I do want to be constructive. But my reasoning leads me to the conclusion that the a priori statement "The game trees arising from board games are a special subclass of the class of all game trees" is not so obviously true: If we look for a precise definition of board game which satisfies the requirement of being compatible with full-fledged chess, Go, Chinese checkers (example of a multi-player board game) and several other popular board games, then one has to realize that it is very difficult to propose restrictions which would not allow almost any game to be satisfy the definition of a board game.]
The question given in the title isn't well posed until we find consensus on a definition of what is a board game. This is actually the main problem, and it turns out to be quite difficult to find a definition which is at the same time somewhat restrictive w.r.t. other non-board games, and nonetheless compatible with those existing games which are considered *board games" by common sense and established dictionaries. To illustrate what I mean, we may notice that even the most "basic" requirements given in earlier answers turned out to be too restrictive:
(1) The finite number of possible constellations: as mentioned somewhere, one could well imagine pieces which grow in (whatever) "strength" without limit. (From the piling up of pieces to denote kings in checkers, its only one step further to imagine a kind of promotion that allows a player to attach an arbitrarily large number to any of his pieces).
(2) A move = piece taken from one location to another, possibly removing another enemy piece: obviously, an arbitrary number of enemy pieces might be removed (as in Go).
Io include moves similar to castling in chess, the definition must potentially allow several pieces from different locations to move at once into possibly any other location.
To include promotions, pieces must be able to be changed into any other of the (as said earlier, possibly infinitely many) pieces.
(3) Also, many of the most popular board games involve throwing dice (which was the case for chess for a long time, not so long ago), asking and answering questions to the other player(s), ....
So, in short, it appears that, in a satisfying definition of a board game,
(a) a "state" of the "board" must comprise more than just a finite number of locations, a possibly variable number of possibly infinitely many distinct pieces (which may be on the board or not), and in addition to the game history, also an arbitrary number of possibly ordered (or possibly infinite) collections of questions and answers.
(b) possible "moves" must include almost arbitrary transitions from one such state to virtually any other state of the board, i.e., in particular, an arbitrary number of pieces can change its location on (or off) the board, in the spirit of Go moves, castling, etc.)
We see that the physical board and the "pieces placed on that board at a given moment" are only a very small part of a possible "state". So it becomes questionable whether restrictions can be found preventing almost any game from being a board game according to a satisfyingly general definition of the latter, i.e., compatible with the most popular board games.
So it is legitimate to wonder whether the notion of "board game" can make a strict logical sense (as opposed to the obvious intuitive sense), unless of course we don't require it to be compatible with real-world chess, Go, Chinese checkers and common popular board games using dice, heaps of questions and/or instructions, etc.