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[(case)-9.6(s)-275.1(with)-280(matching)-286.2(gene)-279.8(trees)-280.8(and)-277.2(species)-283.3(trees.)-281.2(Rosenbe)-9.1(rg)-276.7(\(2007\))-281.7(obtaine)-8.7(d)-273.9(a)-275.7(lower)-281.7(bound)-277(on)-280.4(the)-274.3(ratio)-280.6(of)]TJ
T*
[(the)-268.6(largest)-268(numb)-6.6(er)-267.1(of)-265.3(coalescent)-273.6(histor)-7.6(ies)-264.1(to)-266.1(the)-262.9(smallest)-274.2(number)-267.9(of)-265.3(coalesc)-10.7(ent)-262.9(histor)-7.6(ies)-264.1(at)]TJ
/F14 1 Tf
36.6358 0 TD
(m)Tj
/F1 1 Tf
.9845 0 TD
[(taxa,)-265.8(showing)]TJ
-37.6202 -1.2007 TD
[(that)-289.2(this)-282.9(ratio)-280.6(was)-288(greater)-284.9(than)-286.4(a)-281.4(constant)-289.5(multipl)-8.5(e)-281.4(o)0(f)]TJ
/F7 1 Tf
21.6924 0 TD
(\()Tj
/F8 1 Tf
1.1438 .7227 TD
[()68.9()]TJ
/F15 1 Tf
0 -.7227 TD
(p)Tj
/F10 1 Tf
-.8251 .7227 TD
(p)Tj
/F11 1 Tf
1.3771 -.7227 TD
(=)Tj
/F1 1 Tf
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(32)Tj
/F7 1 Tf
1.0015 0 TD
(\)[\()Tj
/F1 1 Tf
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(5)Tj
/F14 1 Tf
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(m)Tj
/F5 1 Tf
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(-)Tj
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(12)Tj
/F7 1 Tf
1.0015 0 TD
(\))Tj
/F11 1 Tf
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/F7 1 Tf
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/F1 1 Tf
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(4)Tj
/F14 1 Tf
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(m)Tj
/F5 1 Tf
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(6)Tj
/F7 1 Tf
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/F14 1 Tf
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(m)Tj
/F8 1 Tf
1.5478 .7227 TD
[()111.9()]TJ
/F14 1 Tf
0 -.7227 TD
(m)Tj
/F10 1 Tf
-.8251 .7227 TD
(p)Tj
/F1 1 Tf
1.5421 -.7227 TD
[(.)-284.9(Here)-285.7(we)-280.4(improv)-7.1(e)]TJ
-35.6399 -1.195 TD
[(subst)-9(antiall)-8.6(y)-330.8(upon)-338.8(this)-334.1(lower)-338.6(bou)-8(nd,)-332(dem)-7.6(onstratin)-8.7(g)-330.8(that)-340.4(it)-331.4(excee)-11(ds)-332.8(the)-336.9(muc)-7.6(h)-330.8(larger)]TJ
/F7 1 Tf
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([)Tj
/F8 1 Tf
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[()179.4()]TJ
/F14 1 Tf
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(1)Tj
/F10 1 Tf
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(p)Tj
/F11 1 Tf
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(=)Tj
/F7 1 Tf
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(4)Tj
/F8 1 Tf
1.3259 .7227 TD
[()177()]TJ
/F14 1 Tf
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(e)Tj
/F10 1 Tf
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(p)Tj
/F7 1 Tf
1.2747 -.7227 TD
(\)])Tj
/F14 1 Tf
6.9738 0 0 6.9738 481.6062 612.34 Tm
(m)Tj
/F1 1 Tf
9.9626 0 0 9.9626 487.1054 607.918 Tm
(.)Tj
/F2 1 Tf
10.959 0 0 10.959 237.8834 572.0314 Tm
[(2.)-501.9(PRELIMINAR)6.4(IES)]TJ
/F14 1 Tf
-14.9453 -1.6347 TD
[(2.1.)-502(Species)-329.7(trees)-336.4(and)-331.3(coalescent)-327.8(histories)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 84.0755 536.1447 Tm
(A)Tj
/F14 1 Tf
1.1097 0 TD
[(species)-391.4(tre)-8(e)]TJ
/F1 1 Tf
5.2353 0 TD
[(is)-392.4(a)-383.8(bina)-7.6(ry)-384.8(root)-7.8(ed)-390.3(tree)-386.9(equi)-7.6(pped)-391.8(with)-388.1(a)-389.5(labeling)-393.5(for)-393.2(the)-388.1(leaves.)-395.5(As)-387.9(in)-391.3(other)-391.6(studies)-390.6(of)]TJ
-7.3465 -1.2007 TD
[(coal)-9.4(escent)-469.1(histor)-7.6(ies,)-469.3(a)-469.2(single)-468.3(labeling)-473.2(can)-472.5(without)-472.8(loss)-469.1(of)-464.4(general)-9.7(ity)-468.8(be)-464.3(take)-9.4(n)-461.7(a)0(s)-471.1(represe)-10.1(ntative)-472.4(of)-464.4(an)]TJ
0 -1.195 TD
[(un)-7.2(labeled)-403.2(species)-408.5(tree)-404(topolo)-10.1(gy.)-400.2(When)-405.7(the)-399.5(labeli)-10.7(ng)-399.9(is)-398.1(not)-403.4(neede)-9.2(d,)-399.5(we)-399.9(abbre)-9.4(viate)-405.6(the)-399.5(arbitr)-10.2(arily)-400.1(labe)-9.4(led)]TJ
0 -1.2007 TD
[(speci)-11.3(es)-289(tree)-290.2(by)-286.1(its)-293.5(unlabeled)-296.6(shape)-293.1(and)-288.6(consider)-294.4(the)-291.4(labele)-9.8(d)-285.3(and)-288.6(unla)-7.6(beled)-289(topologi)-8.7(es)-289(interc)-10(hangeably)-9.3(.)-284.9(W)0(e)]TJ
T*
[(consi)-9.6(der)-336.1(matching)-343.1(gene)-336.7(trees)-337.7(and)-339.8(species)-340.2(trees)-343.4(with)-336.9(the)-336.9(same)-335.7(labe)-9.4(led)-336.9(topology)]TJ
/F14 1 Tf
32.7605 0 TD
(t)Tj
/F1 1 Tf
.2845 0 TD
(.)Tj
-32.0435 -1.2007 TD
[(We)-370.1(term)-371.6(a)-372.4(coalesc)-10.7(ent)-365.3(histor)-7.6(y)-370.7(for)-364.7(the)-371(case)-373.8(whe)-8(n)-365(the)-371(gene)-370.8(tree)-375.6(and)-368.3(species)-374.3(tree)-375.6(have)-370.8(the)-371(same)-369.8(labele)-9.8(d)]TJ
-1.0015 -1.2007 TD
[(topol)-9.3(ogy)-281.1(a)]TJ
/F14 1 Tf
4.5809 0 TD
[(mat)-9(ching)-281.5(coal)-7.6(escent)-287(history)]TJ
/F1 1 Tf
11.3356 0 TD
[(.)-284.9(Given)-285.4(a)-281.4(species)-294.7(tree)]TJ
/F14 1 Tf
8.974 0 TD
(t)Tj
/F1 1 Tf
.2845 0 TD
[(,)-279.2(a)-281.4(map)-7.6(ping)]TJ
/F14 1 Tf
5.0589 0 TD
(h)Tj
/F1 1 Tf
.7796 0 TD
[(from)-289.4(the)-285.7(internal)-287.4(nodes)-285.6(of)]TJ
/F14 1 Tf
10.8462 0 TD
(t)Tj
/F1 1 Tf
.5577 0 TD
(to)Tj
-42.4174 -1.2007 TD
[(the)-251.5(branc)-9.4(hes)-250(of)]TJ
/F14 1 Tf
6.3791 0 TD
(t)Tj
/F1 1 Tf
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[(is)-250.2(a)-247.2(matching)-257.7(coalescent)-256.6(histor)-7.6(y)-245.5(o)0(f)]TJ
/F14 1 Tf
14.1581 0 TD
(t)Tj
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[(whe)-8(n)-245.5(i)0(t)-246.1(sat)-8(ises)-249(both)-250.5(of)-248.2(the)-251.5(follow)-10.3(ing)-249.7(two)-248(conditi)-9.7(ons:)]TJ
-21.5956 -1.2007 TD
[(\(a\))-264.1(for)-256.6(all)-260(leaves)]TJ
/F14 1 Tf
6.84 0 TD
(x)Tj
/F1 1 Tf
.7056 0 TD
(in)Tj
/F14 1 Tf
1.0357 0 TD
(t)Tj
/F1 1 Tf
.2845 0 TD
.2565 Tc
[(,i)256.5(f)]TJ
/F14 1 Tf
1.3771 0 TD
0 Tc
(x)Tj
/F1 1 Tf
.7056 0 TD
[(descends)-264.2(from)-260.9(intern)-8.2(al)-256.5(node)]TJ
/F14 1 Tf
11.6258 0 TD
(k)Tj
/F1 1 Tf
.7056 0 TD
(in)Tj
/F14 1 Tf
1.0414 0 TD
(t)Tj
/F1 1 Tf
.2788 0 TD
[(,)-256.5(then)]TJ
/F14 1 Tf
2.4981 0 TD
(x)Tj
/F1 1 Tf
.6999 0 TD
[(desce)-10.4(nds)-259.6(from)-260.9(branch)]TJ
/F14 1 Tf
9.0764 0 TD
(h)Tj
/F1 1 Tf
.5008 0 TD
(\()Tj
/F14 1 Tf
.3357 0 TD
(k)Tj
/F1 1 Tf
.4496 0 TD
.2588 Tc
[(\)i)258.8(n)]TJ
/F14 1 Tf
1.6275 0 TD
0 Tc
(t)Tj
/F1 1 Tf
.2845 0 TD
[(;)-253.9(\(b\))-262.3(for)]TJ
-40.0728 -1.2007 TD
[(all)-362.4(int)-7.8(ernal)-364.9(nodes)]TJ
/F14 1 Tf
7.5115 0 TD
(k)Tj
/F1 1 Tf
7.4718 0 0 6.641 153.3543 426.8975 Tm
(1)Tj
9.9626 0 0 9.9626 160.6677 428.5416 Tm
(and)Tj
/F14 1 Tf
1.8096 0 TD
(k)Tj
/F1 1 Tf
7.4718 0 0 6.641 183.1747 426.8975 Tm
(2)Tj
9.9626 0 0 9.9626 190.4881 428.5416 Tm
(in)Tj
/F14 1 Tf
1.1438 0 TD
(t)Tj
/F1 1 Tf
.2845 0 TD
.3589 Tc
[(,i)358.9(f)]TJ
/F14 1 Tf
1.582 0 TD
0 Tc
(k)Tj
/F1 1 Tf
7.4718 0 0 6.641 224.9574 426.8975 Tm
(2)Tj
9.9626 0 0 9.9626 232.2708 428.5416 Tm
[(is)-358.3(a)-361.1(desce)-10.4(ndant)-361.2(o)-6.5(f)]TJ
/F14 1 Tf
7.853 0 TD
(k)Tj
/F1 1 Tf
7.4718 0 0 6.641 314.9857 426.8975 Tm
(1)Tj
9.9626 0 0 9.9626 322.2991 428.5416 Tm
(in)Tj
/F14 1 Tf
1.1438 0 TD
(t)Tj
/F1 1 Tf
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[(,)-364.6(then)-360.4(branc)-9.4(h)]TJ
/F14 1 Tf
5.7987 0 TD
(h)Tj
/F1 1 Tf
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(\()Tj
/F14 1 Tf
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(k)Tj
/F1 1 Tf
7.4718 0 0 6.641 406.9983 426.8975 Tm
(2)Tj
9.9626 0 0 9.9626 410.7401 428.5416 Tm
[(\))-361.2(i)0(s)-364(descended)-368(from)-363.3(or)]TJ
-33.7905 -1.195 TD
[(coin)-7.6(cides)-443.8(with)-445(branc)-9.4(h)]TJ
/F14 1 Tf
9.6398 0 TD
(h)Tj
/F1 1 Tf
.5008 0 TD
(\()Tj
/F14 1 Tf
.3357 0 TD
(k)Tj
/F1 1 Tf
7.4718 0 0 6.641 182.948 414.9353 Tm
(1)Tj
9.9626 0 0 9.9626 186.6897 416.6361 Tm
.4409 Tc
[(\)i)440.9(n)]TJ
/F14 1 Tf
1.9974 0 TD
0 Tc
(t)Tj
/F1 1 Tf
.2788 0 TD
[(.)-444.2(Figure)-448.5(2A)-443.6(shows)-442.6(an)-447.2(exampl)-8.7(e)-440.7(o)0(f)-441.7(a)-440.7(matchi)-11.5(ng)-439.7(coalesc)-10.7(ent)-445(history.)-446.9(The)]TJ
-13.5776 -1.2007 TD
[(exam)-10.2(ples)-247.8(in)-249(Figure)-249.3(2B)-250.9(and)-248.8(2C)-245.2(are)-252.5(not)-244.1(matchi)-11.5(ng)-246.2(coalesc)-10.7(ent)-245.8(histor)-7.6(ies;)-250.6(in)-243.3(Figur)-7.8(e)-247.2(2B,)-245.6(condi)-8.4(tion)-247.6(\(a\))-247(is)-250.2(violated,)]TJ
T*
[(and)-339.8(in)-334.3(Figure)-340.4(2C,)-336.7(condition)-341.3(\(b\))-336.3(is)-335.5(violated.)]TJ
/F14 1 Tf
10.959 0 0 10.959 74.0976 368.7873 Tm
[(2.2.)-502(The)-332.3(lodgepole)-327.4(family)-332.1(of)-331.2(species)-336.1(trees)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 84.0755 350.8723 Tm
[(We)-427(focus)-429.9(here)-429.7(on)-422.6(the)-427.9(numb)-6.6(er)-426.4(of)-424.6(matching)-428.4(coal)-9.4(escent)-429.3(histor)-7.6(ies)-423.4(\()]TJ
/F14 1 Tf
27.3488 0 TD
(histories)Tj
/F1 1 Tf
3.8753 0 TD
(or)Tj
/F14 1 Tf
1.2576 0 TD
[(coal)-7.6(escent)-429.3(histories)]TJ
/F1 1 Tf
8.5529 0 TD
(for)Tj
-42.0361 -1.2007 TD
[(short)-9.7(\))-236.1(for)-245.2(a)-235.9(part)-9.6(icular)-243.3(family)-245(of)-242.5(species)-249.2(trees,)-241.3(\()]TJ
/F16 1 Tf
18.5796 0 TD
(l)Tj
/F14 1 Tf
7.4718 0 0 6.641 264.6991 337.2661 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 268.4409 338.9102 Tm
(\))Tj
/F14 1 Tf
7.4718 0 0 6.641 271.7858 337.2661 Tm
(n)Tj
/F5 1 Tf
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()Tj
/F1 1 Tf
.7208 0 TD
(0)Tj
9.9626 0 0 9.9626 285.9023 338.9102 Tm
[(,)-239.4(that)-243.7(we)-240.6(call)-245.5(the)]TJ
/F14 1 Tf
6.8173 0 TD
[(lodge)-8.4(pole)]TJ
/F1 1 Tf
4.2053 0 TD
[(family.)-245.4(We)-244.9(dene)]TJ
/F16 1 Tf
7.4489 0 TD
(l)Tj
/F1 1 Tf
7.4718 0 0 6.641 475.4266 337.2661 Tm
(0)Tj
9.9626 0 0 9.9626 481.5495 338.9102 Tm
[(as)-243.5(the)]TJ
-40.898 -1.2007 TD
[(1-t)-7(axon)-260.9(tree.)-262.1(For)]TJ
/F14 1 Tf
6.9937 0 TD
(n)Tj
/F5 1 Tf
.6658 0 TD
()Tj
/F1 1 Tf
.7227 0 TD
[(0,)-257.2(we)-257.6(induc)-8.4(tively)-259.3(dene)]TJ
/F16 1 Tf
9.9243 0 TD
(l)Tj
/F14 1 Tf
7.4718 0 0 6.641 261.9779 325.3038 Tm
(n)Tj
/F5 1 Tf
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(+)Tj
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(1)Tj
9.9626 0 0 9.9626 278.6456 326.9479 Tm
[(by)-257.6(append)-8.2(ing)]TJ
/F16 1 Tf
5.7019 0 TD
(l)Tj
/F14 1 Tf
7.4718 0 0 6.641 340.9511 325.3038 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 347.244 326.9479 Tm
[(and)-260.2(a)-258.6(tree)-261.7(with)-262.9(two)-259.4(leaves)-264.3(\(a)]TJ
/F14 1 Tf
11.7908 0 TD
[(cherr)-9.8(y)]TJ
/F1 1 Tf
2.6233 0 TD
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[(\)t)253.1(o)]TJ
-41.8312 -1.2007 TD
0 Tc
[(a)-304.2(com)-7.6(mon)-301.7(root)-303.7(\(Fig.)-303.3(3\).)-305.5(Note)-307.4(that)-300.6(the)-308.4(tree)]TJ
/F16 1 Tf
17.5041 0 TD
(l)Tj
/F14 1 Tf
7.4718 0 0 6.641 253.9842 313.3416 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 260.7307 314.9857 Tm
(has)Tj
/F14 1 Tf
1.6389 0 TD
(m)Tj
/F5 1 Tf
.8877 0 TD
(=)Tj
/F1 1 Tf
.717 0 TD
(2)Tj
/F14 1 Tf
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(n)Tj
/F5 1 Tf
.6715 0 TD
(+)Tj
/F1 1 Tf
.717 0 TD
[(1)-302.4(rather)-305.1(than)]TJ
/F14 1 Tf
5.4857 0 TD
(n)Tj
/F1 1 Tf
.8024 0 TD
[(taxa;)-303.1(we)-303.2(use)]TJ
/F14 1 Tf
5.3662 0 TD
(n)Tj
/F1 1 Tf
.8024 0 TD
[(to)-300.2(denot)-8.4(e)-298.5(the)]TJ
/F14 1 Tf
5.5938 0 TD
(n)Tj
/F1 1 Tf
.5008 0 TD
(th)Tj
-42.4174 -1.2007 TD
[(tre)-8.8(e)]TJ
/F16 1 Tf
1.7584 0 TD
(l)Tj
/F14 1 Tf
7.4718 0 0 6.641 97.1716 301.3795 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 103.3511 303.0235 Tm
[(of)-242.5(the)-251.5(lodgepol)-7.7(e)-247.2(family)-250.7(and)-248.8(perform)-249.9(ou)-7.2(r)-241.7(enum)-8.4(erations)-251.7(accordi)-9.7(ng)-246.2(to)-249(this)-248.8(parameter)-12.9(,)-245.1(later)-248.2(returning)-251.4(to)]TJ
/F14 1 Tf
-2.9363 -1.2007 TD
(m)Tj
/F1 1 Tf
.7227 0 TD
[(,)-336.1(the)-331.2(nu)-7.2(mber)-334.7(of)-333.6(taxa)-9.4(.)-330.4(W)0(e)-341.6(view)]TJ
/F16 1 Tf
12.9972 0 TD
(l)Tj
/F14 1 Tf
7.4718 0 0 6.641 216.2834 289.4172 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 223.37 291.0613 Tm
[(as)-334.6(unlabel)-8.7(ed,)-333.7(or)-333.6(as)-340.2(having)-339.2(an)-333.4(arbitrar)-9.7(y)-336.5(labeling.)]TJ
-13.9817 -1.2007 TD
[(The)-305.5(lodgepol)-7.7(e)-298.5(family)-307.6(\()]TJ
/F16 1 Tf
9.3894 0 TD
(l)Tj
/F14 1 Tf
7.4718 0 0 6.641 183.1747 277.4551 Tm
(n)Tj
/F1 1 Tf
9.9626 0 0 9.9626 186.8598 279.0992 Tm
(\))Tj
/F14 1 Tf
7.4718 0 0 6.641 190.2047 277.4551 Tm
(n)Tj
/F5 1 Tf
.6677 0 TD
()Tj
/F1 1 Tf
.7208 0 TD
(0)Tj
9.9626 0 0 9.9626 207.2692 279.0992 Tm
[(can)-307.5(be)-299.2(seen)-303.7(as)-300.4(a)-298.5(modi)-9.3(cation)-303.9(of)-299.4(the)-302.7(caterpil)-10.3(lar)-299(family)-307.6(of)-299.4(species)-306.1(trees,)-303.9(in)]TJ
-13.3671 -1.195 TD
[(whi)-9(ch)-287.8(a)-287.1(family)-296.2(of)-288(trees)-292.2(is)-290(generated)-297.6(by)-286.1(seque)-8.6(ntially)-291.3(append)-8.1(ing)-289.6(the)-291.4(previous)-292.7(tree)-290.2(in)-288.8(the)-291.4(family)-296.2(and)-288.6(a)-287.1(sing)-7(le)]TJ
0 -1.2007 TD
[(taxo)-7.6(nin)-6.6(stead)-335.7(of)-333.6(a)-338.3(cherry,)-342.5(as)-334.6(in)-334.3(the)-336.9(lodgepol)-7.7(e)-332.6(family)-13.2(to)-334.3(a)-332.6(common)-343.5(root.)]TJ
/F14 1 Tf
10.959 0 0 10.959 74.0976 231.307 Tm
[(2.3.)-502(Dyck)-331(paths)]TJ
/F1 1 Tf
9.9626 0 0 9.9626 84.0755 213.3921 Tm
[(To)-365.5(enumera)-10.5(te)-358.9(histor)-7.6(ies)-366.5(for)-364.7(lodgepol)-7.7(e)-361.1(species)-368.7(trees,)-366.5(we)-365.7(mak)-7.6(e)-361.1(use)-363.8(of)-362(resu)-8(lts)-361.8(that)-363.2(involve)-371.2(certain)-365.3(latt)-7.4(ice)]TJ
-1.0015 -1.2007 TD
[(path)-7.6(s,)-252.7(the)]TJ
/F14 1 Tf
4.1086 0 TD
[(Dyc)-9.8(k)]TJ
/F1 1 Tf
2.3217 0 TD
[(path)-7.6(s)-252.3(\(Stanle)-9.9(y,)-257.2(1999\).)-256.6(A)-255.1(Dyc)-8(k)-251.2(path)-258(of)-259.6(size)]TJ
/F14 1 Tf
17.3619 0 TD
(n)Tj
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