The solution is into $$$3$$$ parts:

1. Making straight chain, so $$$1$$$ node each at a height of $$$0,1,\ldots,p$$$.

2. Adding $$$n-p-2$$$ nodes at height $$$1$$$.

3. Adding the remaining $$$1$$$ node at height which satisfy the score constraint.

Do read the tester's code instead, my code was an alternate lengthy solution.