  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 $s\in[0,1]$, the "tree entropy", to
parametrise the geometry of a halo's entire mass assembly hierarchy, building
on a generalisation 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 $\delta_{\rm
c}/\sigma(M,z)$ 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.

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics;Combinatorics