Analyzing the precision of vegetation height estimation with compact (i.e., single transmit instead of dual transmit) polarimetric interferometric synthetic aperture radar (PolInSAR) with the homogeneous random volume over ground model can help justify the use of this type of radar rather than using the full PolInSAR. However, since compact PolInSAR provides less information than full PolInSAR, a loss of precision in the vegetation height estimation is expected, which can depend on the single transmit polarization. The adaptation of the Cramer-Rao bound (CRB) derived for full PolInSAR in our earlier work to compact PolInSAR measurement provides a general methodology to characterize this loss of precision. Indeed, the CRB is a lower bound of the variance of unbiased estimators that does not depend on the choice of a particular estimation method. We illustrate this methodology for P-band measurements with three synthetic examples chosen for their variability of polarimetric responses. For these examples, it is shown that there can exist a large set of transmit polarizations for which the loss of precision described by the CRB is small (smaller than a factor 2) although there also exist transmit polarizations for which the loss can be large (about a factor 100). This loss of precision is compared with the large dependency of the precision to the vegetation height estimation that can be observed with the vegetation height (more than a factor 100 in the precision described by the CRB) when all the other parameters of the vegetation, ground, and radar system are constant.