Castle Gardens, Malton will be closed to the public until further notice.
The closure comes after a large branch came down from one of the older trees causing damage to fencing and a potential health and safety hazard for visitors to the site.
"Recent strong winds appear to have caused this large branch to fall from the tree damaging the fence, " said Beckie Bennett, Head of Environment for Ryedale District Council.
"Work is currently being undertaken to make the area safe and we hope the site can be re-opened in the near future."
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