Translation:We are afraid of losing that tree.
It is probably a special kind of phrase. Literally: "so that we wouldn't lose the tree." Bát se and equivalents (mít strach, ...) just uses this with abychom.
Compare with: "Bojíme se, že přijdeme o ten strom.", which has the same or almost the same meaning.
Normally, abychom is used to express a purpose, but with these verbs it serves a different purpose.
Děláme to, abychom o ten strom nepřišli. We do that, so that we don't lose the tree.