  A search is performed for electroweak production of a vector-like top quark
partner T of charge 2/3 in association with a top or bottom quark, using
proton-proton collision data at $\sqrt{s} =$ 13 TeV collected by the CMS
experiment at the LHC in 2016. The data sample corresponds to an integrated
luminosity of 35.9 fb$^{-1}$. The search targets T quarks over a wide range of
masses and fractional widths, decaying to a top quark and either a Higgs boson
or a Z boson in fully hadronic final states. The search is performed using two
experimentally distinct signatures that depend on whether or not each quark
from the decays of the top quark, Higgs boson, or Z boson produces an
individual resolved jet. Jet substructure, b tagging, and kinematic variables
are used to identify the top quark and boson jets, and also to suppress the
standard model backgrounds. The data are found to be consistent with the
expected backgrounds. Upper limits at 95% confidence level are set on the cross
sections for T quark-mediated production of tHQq, tZQq, and their sum, where Q
is the associated top or bottom heavy quark and q is another associated quark.
The limits are given for each search signature for various T quark widths up to
30% of the T quark mass, and are between 2 pb and 20 fb for T quark masses in
the range 0.6-2.6 TeV. These results are significantly more sensitive than
prior searches for electroweak single production of T $\to$ tH and represent
the first constraints on T $\to$ tZ using hadronic decays of the Z boson with
this production mode.
