I understand the logic behind the binominal option pricing model, but there seems to me to be slight differences in the calculation. Some add the coupon without discounting, and in other cases we add the coupon but also discount it. I provide two examples from the book below:

On page 288289 example 3 we price a threeyear, annualpay bond with a coupon rate of 5%. At time year 2 the value at the various nodes is 0.5 × [(105/1.08 + 105/1.08)] + 5 = 102.2222. –> The coupon is added but not discounted.

Then on page 330 exhibit 12 there is a ThreeYear 4.25% Annual Coupon Bond Callable at Par One Year and Two Years from Now at 10% Interest Rate Volatility. So this bond has call option. In year one the value is 0,5×(99.738+ 4.25 /1.031681)+(100+ 4.25 )/1.031681) = 100.922. –> The coupon is added and discounted.
The difference is that with a normal bond we seem to add a coupon each year (the coupon that is paid), but this coupon is not discounted. With the embedded option we add the coupon, but we also discount it. Any reason for this difference?