I think the trick for me was realizing there were two functions- first, the discounted value of the call starting from the terminal end point ( t=3). This gives you the expected value of a call from the up case and the down case. Same as for European call.

Except you can also exercise the call at time 2. So look at your current price at time 2. Subtract the strike price, and there’s your payoff.

Now you have two numbers, the discounted expected value of the call exercised at t3 under the risk neutral measure and the payoff if you exercise right now.

We are all greedy bastards in finance, so as greedy bastards, a natural question is - which is bigger? First value, discounted expected payoffs under RN, or second value, pulling the trigger right now?

So t2 value is max of either pull trigger now, or wait. Fix that guy as your T2 value for the “up” case. Now go and do the same for the T2 “down” case. Keep going back until you get to time zero.

Does that help?

Sorry if I wasn't clear in my post. Although I did understand your explanation and it was helpful, but what you're talking about is for the non path dependent American Derivatives.

I'm more specifically looking for help with the path dependent American Derivatives. So what you were basically saying is that

Value of Derivative at time n = max { g(s) , discounted expected value of derivative at (n+1) under RN measure}

where g(s) is the payoff if we exercise the derivative at time n itself.

So I understand this case pretty well. I'm rather not clear about the path dependent case. I'm adding the page from Shreve below:

So I don't follow how is 4.4.1 derived, as in what's the intuition behind it. Specifically, the path dependent case should be a more general case, such that even the non path dependent cases get accounted for in that general formulation.

Another thing that I dont get is, why wouldnt the formulation V[n]= max{ g(s), RN expected value of derivative at n+1} not be valid for all path dependent case too. Like I want to see an example of a derivative (path dependent) for which this formulation, which we use for non path dependent case, doesnt stand valid.

I hope I'm clear in my question. Thanks!