  A recent conjecture of Fyodorov--Hiary--Keating states that the maximum of
the absolute value of the Riemann zeta function on a typical bounded interval
of the critical line is $\exp\{\log \log T -\frac{3}{4}\log \log \log
T+O(1)\}$, for an interval at (large) height $T$. In this paper, we verify the
first two terms in the exponential for a model of the zeta function, which is
essentially a randomized Euler product. The critical element of the proof is
the identification of an approximate tree structure, present also in the actual
zeta function, which allows us to relate the maximum to that of a branching
random walk.
