Pascal's Identity and Trees

Question

Pascal's Identity states that $n \choose k$ = $n-1 \choose k-1$ + $n-1 \choose k$ since if one element is separated from the rest we can claim that either it is chosen (resulting in $k-1$ elements left to choose from) or it is not (resulting in $k$ elements). We can then take either of the two terms in the identity and apply the same reasoning so that it becomes $n \choose k$ = $n-2 \choose k-2$ + $n-2 \choose k-1$ + $n-1 \choose k$. Continuing this reasoning seems to mimic choosing paths in a decision tree with a finite depth that depends entirely on the initial n and k values. Is this something that has been written about or explored in any books on applied combinatorics or graph theory?