We construct continuous (and even invertible) linear operators acting onBanach (even Hilbert) spaces whose restrictions to their respective closedlinear subspaces of chain recurrent vectors are not chain recurrent operators.This construction completely solves in the negative a problem posed by NilsonC. Bernardes Jr. and Alfred Peris on chain recurrence in Linear Dynamics. Inparticular: we show that the non-invertible case can be directly solved viarelatively simple weighted backward shifts acting on certain unrooted directedtrees; then we modify the non-invertible counterexample to address theinvertible case, but falling outside the class of weighted shift operators; andwe finally show that this behaviour cannot be achieved via classical(unilateral neither bilateral) weighted backward sifts (acting on $\mathbb{N}$and $\mathbb{Z}$ respectively) by noticing that a classical shift is a chainrecurrent operator whenever it admits a non-zero chain recurrent vector.