What is the smallest information structure that is required for using the binomial tree to calculate the price of a barrier (up-and-in) option? My gut feeling is any node below the node that reaches the barrier price will be irrelevant.
I'm assuming you're talking about a European option. I did a similar problem for my homework recently, I used the in-out parity for pricing the up and in barrier option.
Basically European Option = Knock up and in Option + Knock up and out option
You can price the up and out easily using Binomial and use BS formula for pricing the European Option, then use the above parity to get the knock up and in.