Legal Question in Real Estate Law in Illinois

Insufficient information. Are the trunks of the trees totally on the neighbor's property, do they straddle the property line? This makes a difference. Also, if the tree branches are healthy but just dropping leaves and sap, this is your problem and always hire a pro because if you kill the tree you could be liable for its replacement! If the branches are dead, the tree owner has a problem because if they fall that owner may be liable for damages, which can include personal injury! And even if you're careful if you interfere with your neighbor's enjoyment of the tree (providing shade) that too can be considered damage you could be liable for. Otherwise the general rule is overhanging branches that are otherwise not causing damage to your property, but are just there, are your problem. Still, the best thing to do is talk to your neighbor and see if you two can reach an accommodation with each other.