  A search is presented for a heavy spin-1 resonance Z' decaying to a top quark
and a vector-like top quark partner T in the lepton+jets final state. The
search is performed using a data set of pp collisions at a centre-of-mass
energy of 13 TeV corresponding to an integrated luminosity of 35.9 fb$^{-1}$ as
recorded by the CMS experiment at the CERN LHC in the year 2016. The analysis
is optimised for final states arising from the T decay modes to a top quark and
a Higgs or Z boson (T $\to$ Ht, Zt). The event selection makes use of resolved
and merged top quark decay products, as well as decays of boosted Higgs bosons
and Z and W bosons using jet substructure techniques. No significant deviation
from the standard model background expectation is observed. Exclusion limits on
the product of the cross section and branching fraction for Z' $\to$ ttT, T
$\to$ Ht, Zt, Wb are presented for various combinations of the Z' resonance
mass and the vector-like T quark mass. These results represent the most
stringent limits to date for the decay mode Z' $\to$ tT $\to$ tHt. In a
benchmark model with extra dimensions, invoking a heavy spin-1 resonance G$^*$,
masses of the G$^*$ between 1.5 and 2.3 TeV and between 2.0 and 2.4 TeV are
excluded for T masses of 1.2 and 1.5 TeV, respectively.
