The generating function of the , 3, component of the Graham TreeLine sequence for ulabeled trees with 1 to, 16, vertices are 3 2 6 4 3 10 7 6 5 4 [1, 1, x, x + x , x + x + x , x + x + x + 2 x + x , 15 11 9 8 7 6 5 x + x + x + 2 x + 2 x + 3 x + x , 21 16 13 12 10 9 8 7 6 28 22 18 x + x + x + 3 x + 3 x + 4 x + 5 x + 4 x + x , x + x + x 17 16 14 13 12 11 10 9 8 7 + 2 x + x + 3 x + 5 x + 2 x + 8 x + 8 x + 9 x + 5 x + x , 36 29 24 23 21 20 19 18 17 16 x + x + x + 2 x + x + x + 3 x + 3 x + 3 x + 2 x 15 14 13 12 11 10 9 8 45 + 10 x + 10 x + 8 x + 19 x + 16 x + 17 x + 7 x + x , x 37 31 30 27 25 24 22 21 20 19 + x + x + 2 x + x + 4 x + 3 x + 3 x + 4 x + 8 x + 8 x 18 17 16 15 14 13 12 11 + 9 x + 8 x + 25 x + 21 x + 28 x + 39 x + 33 x + 27 x 10 9 55 46 39 38 34 32 31 30 28 + 8 x + x , x + x + x + 2 x + x + 3 x + 4 x + x + 3 x 27 26 25 24 23 22 21 20 + 2 x + 11 x + 5 x + 3 x + 10 x + 15 x + 17 x + 22 x 19 18 17 16 15 14 13 12 + 25 x + 34 x + 62 x + 51 x + 75 x + 82 x + 66 x + 43 x 11 10 66 56 48 47 42 40 39 38 36 + 10 x + x , x + x + x + 2 x + x + 3 x + 3 x + x + x 35 34 33 32 31 30 29 28 27 + 3 x + 2 x + 8 x + 8 x + 2 x + 3 x + 8 x + 11 x + 27 x 26 25 24 23 22 21 20 19 + 9 x + 15 x + 30 x + 46 x + 43 x + 68 x + 69 x + 105 x 18 17 16 15 14 13 12 11 + 145 x + 134 x + 185 x + 166 x + 124 x + 63 x + 12 x + x , 78 67 58 57 51 49 48 46 43 42 41 x + x + x + 2 x + x + 3 x + 3 x + x + 4 x + 3 x + 8 x 40 39 37 36 35 34 33 32 31 + 5 x + 3 x + 6 x + 8 x + 8 x + 20 x + 19 x + 5 x + 18 x 30 29 28 27 26 25 24 23 + 23 x + 43 x + 60 x + 28 x + 59 x + 95 x + 126 x + 120 x 22 21 20 19 18 17 16 + 193 x + 198 x + 310 x + 343 x + 358 x + 416 x + 336 x 15 14 13 12 91 79 69 68 61 59 + 224 x + 92 x + 14 x + x , x + x + x + 2 x + x + 3 x 58 55 52 51 50 49 47 45 44 + 3 x + x + 3 x + 3 x + 8 x + 6 x + 3 x + 3 x + 11 x 43 42 41 40 39 38 37 36 + 10 x + 17 x + 10 x + 10 x + 3 x + 23 x + 25 x + 30 x 35 34 33 32 31 30 29 28 + 51 x + 45 x + 19 x + 68 x + 70 x + 136 x + 138 x + 106 x 27 26 25 24 23 22 21 + 189 x + 292 x + 330 x + 373 x + 535 x + 574 x + 823 x 20 19 18 17 16 15 14 13 + 829 x + 903 x + 899 x + 656 x + 383 x + 127 x + 16 x + x 105 92 81 80 72 70 69 65 62 61 , x + x + x + 2 x + x + 3 x + 3 x + x + 3 x + 2 x 60 59 57 56 54 53 52 51 50 + 9 x + 5 x + x + 4 x + 3 x + 8 x + 11 x + 17 x + 10 x 49 48 46 45 44 43 42 41 + 3 x + 10 x + 18 x + 32 x + 36 x + 34 x + 29 x + 28 x 40 39 38 37 36 35 34 33 + 26 x + 71 x + 73 x + 91 x + 138 x + 110 x + 95 x + 220 x 32 31 30 29 28 27 26 + 232 x + 388 x + 346 x + 389 x + 595 x + 863 x + 884 x 25 24 23 22 21 20 + 1142 x + 1454 x + 1648 x + 2112 x + 2039 x + 2168 x 19 18 17 16 15 14 + 1871 x + 1262 x + 633 x + 174 x + 19 x + x ] and in Maple notation [1, 1, x, x^3+x^2, x^6+x^4+x^3, x^10+x^7+x^6+2*x^5+x^4, x^15+x^11+x^9+2*x^8+2*x ^7+3*x^6+x^5, x^21+x^16+x^13+3*x^12+3*x^10+4*x^9+5*x^8+4*x^7+x^6, x^28+x^22+x^ 18+2*x^17+x^16+3*x^14+5*x^13+2*x^12+8*x^11+8*x^10+9*x^9+5*x^8+x^7, x^36+x^29+x^ 24+2*x^23+x^21+x^20+3*x^19+3*x^18+3*x^17+2*x^16+10*x^15+10*x^14+8*x^13+19*x^12+ 16*x^11+17*x^10+7*x^9+x^8, x^45+x^37+x^31+2*x^30+x^27+4*x^25+3*x^24+3*x^22+4*x^ 21+8*x^20+8*x^19+9*x^18+8*x^17+25*x^16+21*x^15+28*x^14+39*x^13+33*x^12+27*x^11+ 8*x^10+x^9, x^55+x^46+x^39+2*x^38+x^34+3*x^32+4*x^31+x^30+3*x^28+2*x^27+11*x^26 +5*x^25+3*x^24+10*x^23+15*x^22+17*x^21+22*x^20+25*x^19+34*x^18+62*x^17+51*x^16+ 75*x^15+82*x^14+66*x^13+43*x^12+10*x^11+x^10, x^66+x^56+x^48+2*x^47+x^42+3*x^40 +3*x^39+x^38+x^36+3*x^35+2*x^34+8*x^33+8*x^32+2*x^31+3*x^30+8*x^29+11*x^28+27*x ^27+9*x^26+15*x^25+30*x^24+46*x^23+43*x^22+68*x^21+69*x^20+105*x^19+145*x^18+ 134*x^17+185*x^16+166*x^15+124*x^14+63*x^13+12*x^12+x^11, x^78+x^67+x^58+2*x^57 +x^51+3*x^49+3*x^48+x^46+4*x^43+3*x^42+8*x^41+5*x^40+3*x^39+6*x^37+8*x^36+8*x^ 35+20*x^34+19*x^33+5*x^32+18*x^31+23*x^30+43*x^29+60*x^28+28*x^27+59*x^26+95*x^ 25+126*x^24+120*x^23+193*x^22+198*x^21+310*x^20+343*x^19+358*x^18+416*x^17+336* x^16+224*x^15+92*x^14+14*x^13+x^12, x^91+x^79+x^69+2*x^68+x^61+3*x^59+3*x^58+x^ 55+3*x^52+3*x^51+8*x^50+6*x^49+3*x^47+3*x^45+11*x^44+10*x^43+17*x^42+10*x^41+10 *x^40+3*x^39+23*x^38+25*x^37+30*x^36+51*x^35+45*x^34+19*x^33+68*x^32+70*x^31+ 136*x^30+138*x^29+106*x^28+189*x^27+292*x^26+330*x^25+373*x^24+535*x^23+574*x^ 22+823*x^21+829*x^20+903*x^19+899*x^18+656*x^17+383*x^16+127*x^15+16*x^14+x^13, x^105+x^92+x^81+2*x^80+x^72+3*x^70+3*x^69+x^65+3*x^62+2*x^61+9*x^60+5*x^59+x^57 +4*x^56+3*x^54+8*x^53+11*x^52+17*x^51+10*x^50+3*x^49+10*x^48+18*x^46+32*x^45+36 *x^44+34*x^43+29*x^42+28*x^41+26*x^40+71*x^39+73*x^38+91*x^37+138*x^36+110*x^35 +95*x^34+220*x^33+232*x^32+388*x^31+346*x^30+389*x^29+595*x^28+863*x^27+884*x^ 26+1142*x^25+1454*x^24+1648*x^23+2112*x^22+2039*x^21+2168*x^20+1871*x^19+1262*x ^18+633*x^17+174*x^16+19*x^15+x^14] ------------------- This took, 643.478, seconds `