  We studied TbTe$_{3}$ using scanning tunneling microscopy (STM) in the
temperature range of 298 - 355 K. As seen in previous STM measurements on
RTe$_{3}$ compounds, our measurements detect a unidirectional charge density
wave state (CDW) in the surface Te-layer with a wavevector consistent with that
of the bulk, q$_{cdw}$ = 0.30 $\pm$ 0.01c$^{*}$. However, unlike previous STM
measurements, and differing from measurements probing the bulk, we detect two
perpendicular orientations for the unidirectional CDWs with no directional
preference for the in-plane crystal axes (a- or c-axis) and no noticeable
difference in wavevector magnitude. In addition, we find regions in which the
bidirectional CDW states coexist. We propose that observation of two CDW states
indicates a decoupling of the surface Te-layer from the rare-earth block layer
below, and that strain variations in the Te surface layer drive the local CDW
direction to the specific unidirectional or, in rare occurrences, bidirectional
CDW orders observed. This indicates that similar driving mechanisms for CDW
formation in the bulk, where anisotropic lattice strain energy is important,
are at play at the surface. In our bias-dependent measurements, we find no
contrast inversion for the CDW state between occupied and empty states. This
finding differs from other quasi 2-dimensional materials containing a hidden
1-dimensional character which leads to a favorable Fermi surface nesting
scenario. Our temperature-dependent measurements provide evidence for localized
CDW formation above the bulk transition temperature, T$_{cdw}$.
