The s-dimensional Hausdorff measure of many randomly constructed sets is almost surely 0 for the ‘correct' exponent s. For stochastic self-similar sets this follows easily by considering the most natural coverings and some argument involving positive variance of the random Hutchinson-Moran sum. Instead, research focussed on modifying the definition of the Hausdorff measure by finding gauge functions under which one obtains positive and finite measure. In this talk I will give some background on the random recursive case and then consider the model of random code-trees with necks which include the random homogeneous and V-variable model.