One of the best HEURISTICS to improve an existing phylogenetic tree is 5-OPTIM.

5-OPTIM rearranges a tree of the form:

There are 15 substantially different ways of rearranging the 5 subtrees (internal nodes are not labelled and not relevant). If the least squares fit for one configuration is known, all the edge lengths and the values of the Least Squares can be computed for all configurations (without examining the insides of the subtrees). The best of the 15 trees is chosen and the procedure is repeated for another central node (the confluence of (e,f,g) until no more improvements can be obtained.