  The simplest Tree-Tensor-States (TTS) respecting the Parity and the
Time-Reversal symmetries are studied in order to describe the ground states of
Long-Ranged Quantum Spin Chains with or without disorder. Explicit formulas are
given for the one-point and two-point reduced density matrices that allow to
compute any one-spin and two-spin observable. For Hamiltonians containing only
one-body and two-body contributions, the energy of the TTS can be then
evaluated and minimized in order to obtain the optimal parameters of the TTS.
This variational optimization of the TTS parameters is compared with the
traditional block-spin renormalization procedure based on the diagonalization
of some intra-block renormalized Hamiltonian.
