  We present a class of exactly solvable tri-junctions of one- and
two-dimensional spin systems. Based on the geometric criterion for solvability,
we clarify the sufficient condition for the junctions so that the spin
Hamiltonian becomes equivalent to Majorana quadratic forms. Then we examine
spin tri-junctions using the obtained solvable models. We consider the
transverse magnetic field Ising spin chains and reveal how Majorana zero modes
appear at the tri-junctions of the chains. Local terms of the tri-junction
crucially affect the appearance of Majorna zero modes, and the tri-junction may
support Majorana zero mode even if the bulk spin chains do not have Majorana
end states. We also examine tri-junctions of two-dimensional SO(5)-spin
lattices and discuss Majorana fermions along the junctions.
