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\section{Introduction}  The main use of plantation \textit{Eucalyptus} species is the production of biomass for the pulp and paper, and bioenergy industries. These trees are fast-growing and can potentially produce high-quality timber for appearance, structural and engineered wood products. Unfortunately, this potential is hindered by the frequent presence of large growth-strains, which are associated with log splitting, warp, collapse and brittleheart, imposing substantial costs on processing \citep{yamamoto2007slides}. \cite{yamamoto2007slides}.  A few technological mitigation strategies have been developed to reduce the incidence of wood defects caused by growth-strain, but they are costly and only partially effective \citep{yamamoto2007slides}. \cite{yamamoto2007slides}.  An alternative approach to the problem is to rely on the genetic control of growth-strain---shown in this article to be highly heritable---to select and grow individuals with low growth-strain. However, measuring growth strains in large numbers of trees (as needed for a successful breeding programme) has been difficult, time consuming and expensive until now. As an example, the largest reported studies to date assessed only 164 \citep{Murphy_2005} \cite{Murphy_2005}  and 216 \citep{naranjo2012early} \cite{naranjo2012early}  trees respectively. The University of Canterbury has developed and implemented a rapid growth-strain testing procedure, based on the work by \citet{Chauhan_2010} \cite{Chauhan_2010}  and \citet{Entwistle_2014}. \cite{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 \citep[e.g.]{senn2012ghosts} [e.g.]\cite{senn2012ghosts}  and in this article we use a Bayesian framework to simulate the missing data from known data, reducing the error induced by zero inflation. 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.