Your last post said it hits "zero at the barrier", which suggested you were talking about a different contract (a KO call). This made me jump in. If it wasn't for that wrong turn, I would have followed along ...

Even the OP seemed confused on this!

Thank you Paul and Alan, very helpful.

@Paul, it wasn't clear to me what you meant all along by saying "plot the value". I thought you were suggesting looking at the values at various nodes of the binomial relative to the exercise payoff at that node. Did you mean plotting the euro-style capped call value at a given point in time (today) as a function of the underlying stock price, where the option value is obtained using the analytical closed-form formula such as the one in Alan's article? I'd like to know if that's what you had in mind, because I will give it a try even if I already have an answer...

@both, How much does the early exercise boundary Bc=L(no dividends) depend on the capped call being traded/registered? And in an even more extreme case, what if the capped call is not only not registered for trading, but there is an explicit sale restriction preventing the holder from selling the derivative to a third party (e.g., the holder has to pay 20% "commission/fee" if they sell to a third party)? 

For every node in the lattice, you'd compare Hold vs. Exercise, but normally the value for Hold is what you would be able to get by selling the option in the open market, so Sell=Hold>Exrcise => don't exercise early (again, no dividends here). What if, as in my example above, Sell<Hold because of some contractual restriction - would it be appropriate to compare Sell vs. Exercise, essentially pushing the "optimal" exercise boundary to some level Bc<L?