OAS and Benchmark Volatility

If you have a callable bond and interest rate volatility in the binomial tree increases (meaning the call option on the bond is worth more and the value of the callable bond is lower) then why would the OAS decrease? Wouldn’t a higher spread over the risk free rates be needed to force the valuation in the tree to equal the bonds mkt price (as the callable bond is worth less)?

Similarly, I would think you would need a decrease in the OAS (discounting at a lower rate) to force a putable bonds valuation to equal its higher price from interest rate volatility (the put option on the bond is worth more).

Can anyone explain this/help me figure out where my logic is off? TIA.

You should use the search function for this; it’s been the subject of quite a few threads this seaason.