Explain the effect of shifts in the yield curve's level and shape on what a callable or putable bond is worth

Rates matter to embedded options through their level and the shape of the curve. As rates fall, a call moves into the money, so the callable bond's price rises less than the straight bond's; as rates rise, a put gains value and supports the putable bond's price. Shape matters through forward rates. An upward-sloping curve implies higher future rates, making calls less likely to be exercised and puts more likely, so a flatter or inverted curve with the same short rate raises the value of a call and lowers the value of a put. Do not look only at the current short rate: the whole curve drives the tree.