  The dynamics of monitored systems can exhibit a measurement-induced phase
transition (MIPT) between entangling and disentangling phases, tuned by the
measurement rate. When the dynamics obeys a continuous symmetry, the entangling
phase further splits into a fuzzy phase and a sharp phase based on the scaling
of fluctuations of the symmetry charge. While the sharpening transition for
Abelian symmetries is well understood analytically, no such understanding
exists for the non- Abelian case. In this work, building on a recent analytical
solution of the MIPT on tree-like circuit architectures (where qubits are
repatedly added or removed from the system in a recursive pattern), we study
entanglement and sharpening transitions in monitored dynamical quantum trees
obeying U (1) and SU (2) symmetries. The recursive structure of tree tensor
networks enables powerful analytical and numerical methods to determine the
phase diagrams in both cases. In the U (1) case, we analytically derive a
Fisher-KPP-like differential equation that allows us to locate the critical
point and identify its properties. We find that the entanglement/purification
and sharpening transitions generically occur at distinct measurement rates. In
the SU (2) case, we find that the fuzzy phase is generic, and a sharp phase is
possible only in the limit of maximal measurement rate. In this limit, we
analytically solve the boundaries separating the fuzzy and sharp phases, and
find them to be in agreement with exact numerical simulations.
