  We consider the global dynamics of finite energy solutions to energy-critical
equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation
(NLH). It is known that any finite energy equivariant solutions to (HMHF)
decompose into finitely many harmonic maps (bubbles) separated by scales and a
body map, as approaching to the maximal time of existence. Our main result for
(HMHF) gives a complete classification of their dynamics for equivariance
indices $D\geq3$; (i) they exist globally in time, (ii) the number of bubbles
and signs are determined by the energy class of the initial data, and (iii) the
scales of bubbles are asymptotically given by a universal sequence of rates up
to scaling symmetry. In parallel, we also obtain a complete classification of
$\dot{H}^{1}$-bounded radial solutions to (NLH) in dimensions $N\geq7$,
building upon soliton resolution for such solutions. To our knowledge, this
provides the first rigorous classification of bubble tree dynamics within
symmetry. We introduce a new approach based on the energy method that does not
rely on maximum principle. The key ingredient of the proof is a monotonicity
estimate near any bubble tree configurations, which in turn requires a delicate
construction of modified multi-bubble profiles also.
