Star formation is a complex process involving several physical mechanisms that interact with each other at different spatial scales. One way to shed some light on this process is to analyse the relation between the spatial distributions of gas and newly formed stars. In order to obtain robust results, it is necessary for this comparison to be made using quantitative and consistent descriptors that are applied to the same star-forming region. We used fractal analysis to characterise and compare in a self-consistent way the structure of the cloud and the distribution of young stellar objects (YSO) in the Dragonfish star-forming complex. Different emission maps of the Dragonfish nebula were retrieved from the NASA/IPAC Infrared Science and the Planck Legacy archives. Moreover, we used photometric information from the AllWISE catalogue to select a total of 1082 YSOs in the region. We derived the physical properties for some of these from their spectral energy distributions (SEDs). For the cloud images and YSOs, the three-dimensional fractal dimension ($D_f$) was calculated using previously developed and calibrated algorithms. The fractal dimension of the Dragonfish nebula ($D_f = 2.6-2.7$) agrees very well with values previously obtained for the Orion, Ophiuchus, and Perseus clouds. On the other hand, YSOs exhibit a significantly lower value on average ($D_f = 1.9-2.0$), which indicates that their structure is far more clumpy than the material from which they formed. Younger Class I and Class II sources have lower values ($D_f = 1.7 0.1$) than more evolved transition disk objects ($D_f = 2.2 0.1$), which shows a certain evolutionary effect according to which an initially clumpy structure tends to gradually disappear over time. The structure of the Dragonfish complex is similar to that of other molecular clouds in the Galaxy. However, we found clear and direct evidence that the clustering degree of the newly born stars is significantly higher than that of the parent cloud from which they formed. The physical mechanism behind this behaviour is still not clear.