  A search is presented for single production of a vector-like quark (T)
decaying to a Z boson and a top quark, with the Z boson decaying leptonically
and the top quark decaying hadronically. The search uses data collected by the
CMS experiment in proton-proton collisions at a center-of-mass energy of 13 TeV
in 2016, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The
presence of forward jets is a particular characteristic of single production of
vector-like quarks that is used in the analysis. For the first time, different
T quark width hypotheses are studied, from negligibly small to 30% of the new
particle mass. At the 95% confidence level, the product of cross section and
branching fraction is excluded above values in the range 0.26-0.04 pb for T
quark masses in the range 0.7-1.7 TeV, assuming a negligible width. A similar
sensitivity is observed for widths of up to 30% of the T quark mass. The
production of a heavy Z' boson decaying to Tt, with T $\rightarrow$ tZ, is also
searched for, and limits on the product of cross section and branching
fractions for this process are set between 0.13 and 0.06 pb for Z' boson masses
in the range from 1.5 to 2.5 TeV.
