  Frustration-free (FF) spin chains have a property that their ground state
minimizes all individual terms in the chain Hamiltonian. We ask how entangled
the ground state of a FF quantum spin-s chain with nearest-neighbor
interactions can be for small values of s. While FF spin-1/2 chains are known
to have unentangled ground states, the case s=1 remains less explored. We
propose the first example of a FF translation-invariant spin-1 chain that has a
unique highly entangled ground state and exhibits some signatures of a critical
behavior. The ground state can be viewed as the uniform superposition of
balanced strings of left and right parentheses separated by empty spaces.
Entanglement entropy of one half of the chain scales as log(n)/2 + O(1), where
n is the number of spins. We prove that the energy gap above the ground state
is polynomial in 1/n. The proof relies on a new result concerning statistics of
Dyck paths which might be of independent interest.
