According to the Online Encyclopaedia of Integer Sequences (OEIS) for n 
= 6,7,8,9,10 leaves there are f(n) = 2,2,4,6,11 unrooted, unlabelled 
binary trees with n leaves, see https://oeis.org/A129860.

Hence, if for each n we can identify f(n) trees that generate distinct 
spectra, it follows that in the space of unrooted, unlabelled binary 
trees with n leaves the g_k spectrum DOES completely identify the tree 
topology (because two isomorphic trees obviously generate the same 
spectrum).

The trees shown below show exactly this. (Note that for n=2,3,4,5 the 
claim is trivial because there is only one unrooted, unlabelled binary 
tree topology on n taxa).

-------------------------
*** 6 taxa
------------------------
(1,(2,(3,(4,(5,6)))));
89, 5, 2, 1, 1, 1
((1,2),((3,4),(5,6)));
89, 5, 1, 1, 1, 1
-----
After eliminating duplicates:
89, 5, 1, 1, 1, 1
89, 5, 2, 1, 1, 1
2 spectra seen

-------------------------
*** 7 taxa
------------------------

((1,2),((3,(4,5)),(6,7)));
233, 8, 2, 1, 1, 1, 1
(1,(2,(3,(4,(5,(6,7))))));
233, 8, 3, 1, 1, 1, 1
-----
After eliminating duplicates:
233, 8, 2, 1, 1, 1, 1
233, 8, 3, 1, 1, 1, 1
2 spectra seen


-------------------------
*** 8 taxa
------------------------
(1,(2,(3,(4,(5,(6,(7,8)))))));
610, 13, 4, 2, 1, 1, 1, 1
((1,2),((3,(4,5)),(6,(7,8))));
610, 13, 3, 1, 1, 1, 1, 1
(((1,2),(3,4)),((5,6),(7,8)));
610, 13, 2, 2, 1, 1, 1, 1
((1,2),((3,4),(5,(6,(7,8)))));
610, 13, 3, 2, 1, 1, 1, 1
-----
After eliminating duplicates:
610, 13, 2, 2, 1, 1, 1, 1
610, 13, 3, 1, 1, 1, 1, 1
610, 13, 3, 2, 1, 1, 1, 1
610, 13, 4, 2, 1, 1, 1, 1
4 spectra seen


-------------------------
*** 9 taxa
------------------------
((1,2),(3,(4,(5,(6,(7,(8,9)))))));
1597, 21, 6, 3, 1, 1, 1, 1, 1
(((1,2),(3,4)),(5,(6,(7,(8,9)))));
1597, 21, 4, 3, 1, 1, 1, 1, 1
(((1,2),3),((4,5),(6,(7,(8,9)))));
1597, 21, 5, 2, 1, 1, 1, 1, 1
((((1,2),3),((4,5),6)),(7,(8,9)));
1597, 21, 5, 1, 1, 1, 1, 1, 1
(((1,2),(3,4)),(5,((6,7),(8,9))));
1597, 21, 3, 3, 1, 1, 1, 1, 1
(((1,2),(3,4)),((5,6),(7,(8,9))));
1597, 21, 3, 2, 1, 1, 1, 1, 1
-----
After eliminating duplicates:
1597, 21, 3, 2, 1, 1, 1, 1, 1
1597, 21, 3, 3, 1, 1, 1, 1, 1
1597, 21, 4, 3, 1, 1, 1, 1, 1
1597, 21, 5, 1, 1, 1, 1, 1, 1
1597, 21, 5, 2, 1, 1, 1, 1, 1
1597, 21, 6, 3, 1, 1, 1, 1, 1
6 spectra seen


-------------------------
*** 10 taxa
------------------------
((1,(2,(3,(4,5)))),(6,(7,(8,(9,10)))));
4181, 34, 9, 4, 2, 1, 1, 1, 1, 1
((1,(2,(3,(4,5)))),((6,7),(8,(9,10))));
4181, 34, 7, 3, 2, 1, 1, 1, 1, 1
((1,(2,(3,(4,5)))),(6,((7,8),(9,10))));
4181, 34, 6, 4, 2, 1, 1, 1, 1, 1
((6,((7,8),(9,10))),(1,((2,3),(4,5))));
4181, 34, 4, 4, 2, 1, 1, 1, 1, 1
(((6,7),(8,(9,10))),(1,((2,3),(4,5))));
4181, 34, 5, 3, 2, 1, 1, 1, 1, 1
(((6,7),(8,(9,10))),((1,2),(3,(4,5))));
4181, 34, 5, 2, 2, 1, 1, 1, 1, 1
(((1,2),(3,4)),((5,(6,7)),(8,(9,10))));
4181, 34, 5, 2, 1, 1, 1, 1, 1, 1
(((1,2),(3,4)),((5,6),((7,8),(9,10))));
4181, 34, 3, 3, 1, 1, 1, 1, 1, 1
(((1,2),(3,4)),((5,6),(7,(8,(9,10)))));
4181, 34, 5, 3, 1, 1, 1, 1, 1, 1
((1,(2,(3,4))),((5,6),(7,(8,(9,10)))));
4181, 34, 8, 3, 1, 1, 1, 1, 1, 1
((1,(2,(3,4))),((5,(6,7)),(8,(9,10))));
4181, 34, 8, 2, 1, 1, 1, 1, 1, 1
((1,(2,(3,4))),((5,6),((7,8),(9,10))));
4181, 34, 5, 3, 1, 1, 1, 1, 1, 1
-----
After eliminating duplicates:
4181, 34, 3, 3, 1, 1, 1, 1, 1, 1
4181, 34, 4, 4, 2, 1, 1, 1, 1, 1
4181, 34, 5, 2, 1, 1, 1, 1, 1, 1
4181, 34, 5, 2, 2, 1, 1, 1, 1, 1
4181, 34, 5, 3, 1, 1, 1, 1, 1, 1
4181, 34, 5, 3, 2, 1, 1, 1, 1, 1
4181, 34, 6, 4, 2, 1, 1, 1, 1, 1
4181, 34, 7, 3, 2, 1, 1, 1, 1, 1
4181, 34, 8, 2, 1, 1, 1, 1, 1, 1
4181, 34, 8, 3, 1, 1, 1, 1, 1, 1
4181, 34, 9, 4, 2, 1, 1, 1, 1, 1
11 spectra seen