Hi, Dead trees are probably not a good bet. The wood in most trees degrades quickly after the tree dies. Yew and osage are definitely exceptions! Elms of almost any species will make an excellent bow. Just cut them while still living. Elm green or dry will take a LOT!!!! of work to split.
Locust and osage are both excellent bow woods. There are some people on this forum who see no reason to ever use anything except osage. Black locust has a following too, though the wood will chrysal if the tiller isn't near perfect.
For me the best time to collect bow wood is when it's available. The bark will be difficult to remove if harvested in the winter, but that may not be a problem if you want to season the wood with the bark intact. I mainly harvest white woods, so I'll let people with more expeience in handling locust and osage give you thier secrets to harvesting the perfect stave.
Ron