This isn't a full answer but some setup and initial attempt.

Maybe we can formalize a class of "board games" that we're happy with first. I'd propose that:

• A board is a set of locations.

• A piece occupies a single location on the board at a given time. Probably each piece "belongs" to exactly one player.

• A move consists of placing a new piece on the board at some location, removing one from the board, and/or moving a piece from one location to another. (e.g. capturing in chess does two of these things)

• A history is a list of past moves.

• The "rules" specify, for a given history, which player is to move and which moves are legal at that time.

Now I am pretty happy with this definition, but the problem is that (I'm pretty sure) an arbitrary game tree can be phrased as a board game above, if we allow enough "pieces" and arbitrarily complicated changes to the legal moves given a history.

But I think we get closer with this restriction:

(consistency) the set of legal moves depend only on the board position, not the history.

and even farther with

(simplicity) each piece's set of legal moves depend only on the occupancy status of the board's locations, not the identity of the pieces there.

Of course castling, en passant, and 3-repetition-draw don't satisfy this, so it technically rules out chess. But I think these restrictions make it easier to think about your question.

I'd from here we could say that a board game tree always has each node labeled by the board "position" (set of current pieces and their locations) and history of moves. At each node, its moves are labeled by pairs (piece, location) consisting of pieces that can legally be moved (including those that may be newly placed on the board, as in Go) along with a legal destination location. Furthermore if we have the consistency and simplicity conditions, then these do not depend on the history, only the board position, and can be calculated "simply" from that position for each piece.

This isn't a full answer but some setup and initial attempt.

Maybe we can formalize a class of "board games" that we're happy with first. I'd propose that:

• A board is a set of locations.

• A piece occupies a single location on the board at a given time. Probably each piece "belongs" to exactly one player.

• A move consists of placing a new piece on the board at some location, removing one from the board, and/or moving a piece from one location to another. (e.g. capturing in chess does two of these things)

• A history is a list of past moves.

• The "rules" specify, for a given history, which player is to move and which moves are legal at that time.

Now I am pretty happy with this definition, but the problem is (as has been noted) an arbitrary game tree can be phrased as a board game in this way.

But I think we get closer with this restriction:

(consistency) the set of legal moves depend only on the board position, not the history.

and even farther with

(simplicity) each piece's set of legal moves depend only on the occupancy status of the board's locations, not the identity of the pieces there.

Edit: I think we can come up with some better "simplicity" condition having to do with a simple set of rules for each piece, but formalizing it seems tricky to me.

Of course castling, en passant, and 3-repetition-draw don't satisfy this (edit: nor does e.g. a pin of a piece to the king), so it technically rules out chess. But I think these restrictions make it easier to think about your question.

From here we could say that a board game tree always has each node labeled by the board "position" (set of current pieces and their locations) and history of moves. At each node, its moves are labeled by pairs (piece, location) consisting of pieces that can legally be moved (including those that may be newly placed on the board, as in Go) along with a legal destination location. Furthermore if we have the consistency and simplicity conditions, then these do not depend on the history, only the board position, and can be calculated "simply" from that position for each piece.