Bond coupon being paid, yet bond was called...

This a really stupid question and I hesitate to ask because the answer will likely be obvious (once I read your responses; and thank you in advance by the way):

When valuing a callable bond in the binomial tree, and the option is exercised at a particular node…we add the coupon payment to the strike price of the option (say, par or 100 plus a coupon payment) and continue on through the tree. But why? The bond has been called; the issuer has it back. I thought with that transaction, they get the bond and I get my 100/par and go on my merry little way,

But in the tree we treat it as if I deliver the bond AND I get the coupon.

It would appear that we assume the coupon gets paid, even if they call it. Is that reality? If not, why do we add the coupon?

Interest is paid in arrears. The bondholder has earned it, so the issuer pays it.

Thank you!

You’re welcome.