When should i cut down my lilac tree???

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Ah, that's really up to you. If it's more than 50% dead, it's probably a goner. If it's dead or nearly so, any time you want to cut it down, fine with me. I wouldn't wait too long because lilac wood is tough to cut when alive, murder when dead.

There's no season for removing dead material..it's dead. Actually if it died because of insects, say the lilac borer, getting it out of the yard and to the proper land disposal as fast as possible might save other lilac trees in your neighborhood.