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 $s=0.3$ for first
generation haloes. This distribution exhibits weak dependences on halo mass $M$
and redshift $z$, which can be reduced to a single dependence of $\langle
s\rangle$ 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.