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