The willow tree is a bit of a sore subject when it comes to degus. Many German degu owners are happy to use willow wood with their degus, but English users (partly due to Degutopia, of course) disagree and say the wood is toxic. Please read the arguments for and against (coming soon) before deciding for yourself about this.
Willow wood is reported to result in respiratory infections, usually evident by a sneezing degu. While cute, it isn't a good thing to hear, however if you're using willow in your cage and your degu sneezes it doesn't mean the wood is to blame...maybe you use wood shavings that are "dust extracted" but the manufacturers didn't do a good enough job of it, or maybe the room they're in is dusty. If you notice sneezes, remove the wood from their cage and monitor them for a while to see if it continues. If it stops, consider reintroducing the wood to see if it starts again (and obviously removing it permanently if it does) so then you know it definitely is the wood causing the problem.
It should be noted that this problem probably doesn't affect every degu, some might be more susceptible to it than others.
The mistake that most sites seem to have made is the kind of willow that is safe. However, Chinchillas2home states: "WHITE willow - we all know this is used for Aspirin - If willow is to be avoided then it's the white! There are so many varieties and all would be related - but white contains 13% tannin as its chief constituent, also a small quantity of salicin, the other willows (mainly reed and bushes that are used for basket making and such) would contain a lesser amount, and most pets chew and hack through weeping and bush/reed willow products without any harm done" so there you have it. If you have weeping willow trees nearby then you can grab the branches and weave stuff for your degu...just follow the Wood preparation guidelines first!