The library re-opened on Saturday following Friday's closure but the tree did cut off the telephone link to the library and the new, improved electronic connection.
Helen Compton, the council's libraries ICT and lifelong learning officer, said everything had been done to keep disruption to library users to a minimum.
She said, "We were able to provide a basic service from Haxby library and managed to provide telephone information from Acomb library by re-directing the telephone line. People wanting to check information on loans or make reservations using the on-line service were able to contact any other library.
"Through our new service support for the electronic connection we were able to get BT along yesterday and thankfully they managed to put everything right by the afternoon. We would like to thank everyone for their patience over the last few days."
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