Do the object left in radley tree have any connection with the condition in which jem find his trousers?

Yes, there is a connection. The object in the Radley tree, which is a pair of carved soap figures resembling Jem and Scout, is symbolic of Boo Radley reaching out to the children. Jem finding his torn trousers after leaving them caught on the Radley fence reflects Boo's protective nature towards them. Both instances highlight Boo's caring presence in their lives.