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[Edit: Two quick downvotes let me think that my answer is 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. We see here that it is highly nontrivial to find his is far from being the real question/problem We must focus on this, and the latter appears to be quite arbitrary and unsatisfying whatever restriction(s) would be chosen. To illustrate what I mean, consider that even the most "basic" requirements given in earlier answers are unsatisfying:

(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.

[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:

(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.

Indeed, the question given in the title does not make sense without the prior definition of what is a "board game", and the latter appears to be quite arbitrary and unsatisfying whatever restriction(s) would be chosen. To illustrate what I mean, consider that even the most "basic" requirements given in earlier answers are unsatisfying:  (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).

• Also, to 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 (a) a "state" of the "board" most comprise more than a finite number of locations, possibly infinite number of possible pieces, and other accessories like dice, clocks, infinite collections of questions and answers ... (b) a transition from one such state to virtually any other state of the board, must be compatible with a general definition. We see that the physical board and the "currently active pieces placed on that board" are only an infinitesimal part of a possible "state". So, whatever restriction could prevent any game of being a board game, according to any satisfyingly general definition of the latter (i.e., compatible with the most popular board games) ? We should admit that on a closer look, the "board" itself is, in spite of the first appearance, an (almost(?)) negligible part of most board games. And any non board game could probably be a board game with a possibly trivial (empty or singleton) set of "locations". So I 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.

[Edit: Also, multi-player games must be comprised in a satisfying definition, as to include the very classical board games of Halma and Chinese checkers.]

[Edit: Two quick downvotes let me think that my answer is 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. We see here that it is highly nontrivial to find his is far from being the real question/problem We must focus on this, and the latter appears to be quite arbitrary and unsatisfying whatever restriction(s) would be chosen. To illustrate what I mean, consider that even the most "basic" requirements given in earlier answers are unsatisfying: 

(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.

Indeed, the question given in the title does not make sense without the prior definition of what is a "board game", and the latter appears to be quite arbitrary and unsatisfying whatever restriction(s) would be chosen. To illustrate what I mean, consider that even the most "basic" requirements given in earlier answers are unsatisfying:  (1) The finite number of possible constellations: as mentioned somewhere, one could well imagine pieces which grow in (whatever) "strength" without limit (one can 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). Also, to 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. Also, many board games involve throwing dice (which was the case for chess in medieval times), asking and answering questions to the other player, .... So, in short, it appears that any imaginable transition from one "state" of the "board" (comprising much more than the finite number of locations, infinite number of possible pieces, and other accessories like dice, clocks, infinite collections of questions and answers ...) to virtually any other state of the board, must be compatible with a general definition. We see that the physical board and the "currently active pieces placed on that board" are only an infinitesimal part of a possible "state". So, whatever restriction could prevent any game of being a board game, according to any satisfyingly general definition of the latter, which should encompass at least all of the well known classical board games? We should admit that on a closer look, the "board" itself is an (almost(?)) negligible part of many board games. And any non board game would probably be a board game with a possibly trivial (empty or singleton) set of "locations". So I 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. [Edit: Also, multi-player games must be comprised in a satisfying definition, as to include the very classical board games of Halma and Chinese checkers.]

Indeed, the question given in the title does not make sense without the prior definition of what is a "board game", and the latter appears to be quite arbitrary and unsatisfying whatever restriction(s) would be chosen. To illustrate what I mean, consider that even the most "basic" requirements given in earlier answers are unsatisfying:  (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).

• Also, to 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 (a) a "state" of the "board" most comprise more than a finite number of locations, possibly infinite number of possible pieces, and other accessories like dice, clocks, infinite collections of questions and answers ... (b) a transition from one such state to virtually any other state of the board, must be compatible with a general definition. We see that the physical board and the "currently active pieces placed on that board" are only an infinitesimal part of a possible "state". So, whatever restriction could prevent any game of being a board game, according to any satisfyingly general definition of the latter (i.e., compatible with the most popular board games) ? We should admit that on a closer look, the "board" itself is, in spite of the first appearance, an (almost(?)) negligible part of most board games. And any non board game could probably be a board game with a possibly trivial (empty or singleton) set of "locations". So I 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.

[Edit: Also, multi-player games must be comprised in a satisfying definition, as to include the very classical board games of Halma and Chinese checkers.]