From the modified start branch the optimizer follows all 4 adjacent branches and modifies each of them (twice). From each modified adjacent branch the optimizer follows the 3 not yet visited adjacent branches and modifies these. This is repeated until the 'Maximum recursion depth' is reached.
When walking along 15 branches this results in ~ 1.3 * 10^12 possible modifications, far too much to test them all. These possible modifications are called "paths" here.
To reduce the number of paths to follow, two path reductions are available and active by default:

static path reduction
Statically defines how many paths are tested at which depth. At depth=1 the number of allowed paths may be 1 to 8, for greater depths if may be 1 to 6.
The optimizer always follows the best modifications (according to the ´Parsimony value´).
For depths beyond 5, the static path reduction is hardcoded as 1.

dynamic path reduction
Dynamically defines a limit how strong intermediate modifications may worsen the tree.
The 'start costs' and 'maximum costs' are specified as absolute degradation of the parsimony value.
Both path reductions may be used either together or separately.
Without any path reduction a complete recursion through all possible topologies is performed. When doing so, a good starting value for 'Maximum recursion depth' is 8.
