Linking the properties of galaxies to the assembly history of their dark matter haloes is a central aim of galaxy evolution theory. This paper introduces a dimensionless parameter $sin[0,1]$, the tree entropy, to parametrise the geometry of a halos entire mass assembly hierarchy, building on a generalisation of Shannons information entropy. By construction, the minimum entropy ($s=0$) corresponds to smoothly assembled haloes without any mergers. In contrast, the highest entropy ($s=1$) represents haloes grown purely by equal-mass binary mergers. Using simulated merger trees extracted from the cosmological $N$-body simulation SURFS, we compute the natural distribution of $s$, a skewed bell curve peaking near $s=0.4$. This distribution exhibits weak dependences on halo mass $M$ and redshift $z$, which can be reduced to a single dependence on the relative peak height $delta_{rm c}/sigma(M,z)$ in the matter perturbation field. By exploring the correlations between $s$ and global galaxy properties generated by the SHARK semi-analytic model, we find that $s$ contains a significant amount of information on the morphology of galaxies $-$ in fact more information than the spin, concentration and assembly time of the halo. Therefore, the tree entropy provides an information-rich link between galaxies and their dark matter haloes.
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