  Weighted shifts on directed trees are a decade old generalisation of
classical shift operators in the sequence space $\ell^2$. In this paper we
introduce the joint backward extension property (JBEP) for classes of weighted
shifts on directed trees. If a class satisfies JBEP, the existence of a common
backward extension within the class for a family of weighted shifts on rooted
directed trees does not depend on the additional structure of the big tree (of
fixed depth). We decide whether several classes of operators have JBEP. For
subnormal or power hyponormal weighted shifts the property is satisfied, while
it fails for completely hyperexpansive or quasinormal. Nevertheless some
positive results on joint backward extensions of completely hyperexpansive
weighted shifts are proven.
