(root 0,0) >>>>
   
>>>
  
>>
 
>

(leaf n,n)
At each point of the sequence, the number of "(" is no less than that of ")". So all possible states are nodes in the above figure. forward means adding a "(", and downward means adding a ")". Thus, there is a binary tree! with each node represented by a coordinate.
For n pairs, there are (n+2)(n+1)/2 = O(n^2). Can be