Maximum non-intersecting subset of circles
If it were me, I would encode this as an instance of ILP (integer linear programming) and let an off-the-shelf ILP solver deal with the backtracking and pruning.

Introduce $n$ variables $x_1,\dots,x_n$, where $x_i=1$ means that you include the $i$th circle in the subset and $x_i=0$ means you don't include it.  Add the following constraints:


$0 \le x_i \le 1$, for each $i$: this forces each $x_i$ to be either 0 or 1.
$x_i + x_j \le 1$, for each pair $i,j$ of circles that intersect each other: this enforces that you cannot choose two overlapping circles.


Now maximize the value of $x_1+x_2+\dots+x_n$ (this is the objective function).  Ask an ILP solver to find the optimal solution for you.  Internally, the ILP solver will likely use backtracking, pruning, branch-and-bound, and other sophisticated methods.

This will be a very efficient use of your time: rather than trying to program up each of those methods and debug your implementation, you can rely upon an existing well-tested ILP solver.  Also, given the amount of effort that has gone into optimizing ILP solvers, this approach might well perform better than any backtracking algorithm you're likely to come up with.