In section 12/36. Arbitrage free valuation the value of the bond at the certain nodes of the interest rate tree is after we have added the coupon.
In section 13/37 Bonds with embedded options the value is before we add the coupon. This latter makes more sense to me because if we add the coupon than almost all bonds would become callable and they would never become putable.
But how is this right? If a question asks me not the price at time zero but at a certain later node, and it is a straight bond which should be the value? The one with coupon added or not?