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 s ϵ [0, 1], the 'tree entropy', to parametrize the geometry of a halo's entire mass assembly hierarchy, building on a generalization of Shannon's 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 δc/σ(M, z) in the matter perturbation field. By exploring the correlations between s and global galaxy properties generated by the SHARKsemi-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.