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The University of Canterbury has developed and implemented a rapid growth-strain testing procedure, based on the work by \citet{Chauhan_2010} and \citet{Entwistle_2014}. In order to minimise the time taken to measure growth-strain on each tree, the rapid testing procedure does not account for negative values, where the wood in the centre of the stem is under tension rather than compression, assigning instead a zero that results in left-censored datasets.   Left-censored data are common in research areas where detection limits are high compared to the measured values, such as testing for the presents of drugs in an animal. There are many approaches to deal with censoring \cite[e.g.]{senn2012ghosts} and in this article we use a Bayesian framework to simulate impute  the missing data from known data, reducing the error induced by zero inflation. A Bayesian approach makes easier to include model complexity (e.g. censoring) while accounting for the hierarchical nature of the data. In addition, one can easily obtain complex distributions of functions of covariance components, like heritabilities, as a byproduct of the estimation process \cite{cappa2006bayesian}. There are several examples of Bayesian applications in forest genetics; for example: \cite{soria1998application} (univariate analysis of growth traits), \cite{cappa2006bayesian} (multivariate analysis of growth traits) and \cite{apiolaza2011genetic} (multivariate analysis of early wood properties).  We run a pilot study in a couple of \textit{Eucalyptus bosistoana} progeny tests consisting of both seed and coppice grown stems, which included 623 individuals from 40 half sibling families. Our estimates of narrow-sense heritability were obtained from left-censored growth-strain data and other wood properties, utilising a Bayesian approach. These results were used to design a much larger evaluation of the \textit{E. bosistoana} breeding population.