The zeroth-order general Randić index (usually denoted by R 0 α ) and variable sum exdeg index (denoted by SEI a ) of a graph G are defined as R 0, a is a positive real number different from 1 and α is a real number other than 0 and 1. A segment of a tree is a path P , whose terminal vertices are branching or pendent, and all non-terminal vertices (if exist) of P have degree 2. For n ≥ 6, let PT n,n1 , ST n,k , BT n,b be the collections of all n-vertex trees having n 1 pendent vertices, k segments, b branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections PT n,n1 , ST n,k , BT n,b . The obtained extremal trees for the collection ST n,k are also extremal trees for the collection of all n-vertex trees having fixed number of vertices with degree 2 (because it is already known that the number of segments of a tree T can be determined from the number of vertices of T with degree 2 and vise versa).

We show that on any compact Kähler surface obstructions to the existence of solutions to the J-equation, deformed Hermitian-Yang-Mills equation, and Z-critical equation can each be determined using a finite number of effective conditions, where the number of conditions needed is bounded above by the Picard number of the surface. The novel technique is the use of Zariski decomposition, which produces a finite set of 'test curves' uniform across compact sets of initial data. This shows that a finite number of polynomials locally cut out the set of classes for which the respective equations are solvable, giving a first analogue on the PDE side of the locally finite wall-and-chamber decomposition central in the theory of Bridgeland stability. We moreover deduce that the boundary of the set of classes admitting a solution to the dHYM and Z-critical equations form real algebraic sets of codimension one.As an application we characterize optimally destabilizing curves for the J-equation and dHYM equation, and prove a non-existence result for optimally destabilizing test configurations for uniform J-stability. In connection with this we show that on a large class of surfaces one can find polarizations that are J-stable but not uniformly J-stable. We finally remark on an improvement to convergence results for the J-flow, line bundle mean curvature flow, and dHYM flow.