  We obtain an important generalization of the mechanical solution given by S.
Gueron and R. Tessler w.r. to the weighted Fermat-Torricelli problem which
derives a new structure of solutions which may be called oscillatory
Fermat-Torricelli trees. The weighted Fermat-Torricelli problem in R^2 states
that: Given three points in R^2 and a positive real number (weight) which
correspond to each point , find the point (weighted Fermat-Torricelli point)
such that the sum of the weighted distances to these three points is minimized.
By applying the mechanical device of Pick and Polya the oscillatory tree
solution is a new solution w.r to the weighted Fermat-Torricelli problem for a
given isosceles triangle with corresponding two equal weights at the vertices
of the base segment. it is worth mentioning that after time t the oscillatory
knot of the mechanical system passes from the weighted Fermat-Torricelli point
with non zero velocity. Furthermore, we give a numerical example to verify the
structure of an oscillatory Fermat-Torricelli tree for a given isosceles
triangle with equal weights.
