  Entanglement phase transitions in quantum chaotic systems subject to
projective measurements and in random tensor networks have emerged as a new
class of critical points separating phases with different entanglement scaling.
We propose a mean-field theory of such transitions by studying the entanglement
properties of random tree tensor networks. As a function of bond dimension, we
find a phase transition separating area-law from logarithmic scaling of the
entanglement entropy. Using a mapping onto a replica statistical mechanics
model defined on a Cayley tree and the cavity method, we analyze the scaling
properties of such transitions. Our approach provides a tractable,
mean-field-like example of an entanglement transition. We verify our
predictions numerically by computing directly the entanglement of random tree
tensor network states.
