Question

I need 3 pages on Penny Treese

Art Appreciation

Answer 1

Due to Terms and condition (Limitation of text) i can give this much pages only.

Penny iis ia iprogram ithat iwill ifind iall iof ithe imost iparsimonious itrees iimplied iby iyour idata. iIt idoes iso inot iby iexamining iall ipossible itrees, ibut iby iusing ithe imore isophisticated i"branch iand ibound" ialgorithm, ia istandard icomputer iscience isearch istrategy ifirst iapplied ito iphylogenetic iinference iby iHendy iand iPenny i(1982). i(J. iS. iFarris i[personal icommunication, i1975] ihad ialso isuggested ithat ithis istrategy, iwhich iis iwell-known iin icomputer iscience, imight ibe iapplied ito iphylogenies, ibut ihe idid inot ipublish ithis isuggestion).

There iis, ihowever, ia iprice ito ibe ipaid ifor ithe icertainty ithat ione ihas ifound iall imembers iof ithe iset iof imost iparsimonious itrees. iThe iproblem iof ifinding ithese ihas ibeen ishown i(Graham iand iFoulds, i1982; iDay, i1983) ito ibe iNP-complete, iwhich iis iequivalent ito isaying ithat ithere iis ino ifast ialgorithm ithat iis iguaranteed ito isolve ithe iproblem iin iall icases i(for ia idiscussion iof iNP-completeness, isee ithe iScientific iAmerican iarticle iby iLewis iand iPapadimitriou, i1978). iThe iresult iis ithat ithis iprogram, idespite iits ialgorithmic isophistication, iis iVERY iSLOW.

The iprogram ishould ibe islower ithan ithe iother itree-building iprograms iin ithe ipackage, ibut iuseable iup ito iabout iten ispecies. iAbove ithis iit iwill ibog idown irapidly, ibut iexactly iwhen idepends ion ithe idata iand ion ihow imuch icomputer itime iyou ihave. iIT iIS iVERY iIMPORTANT iFOR iYOU iTO iGET iA iFEEL iFOR iHOW iLONG iTHE iPROGRAM iWILL iTAKE iON iYOUR iDATA. iThis ican ibe idone iby irunning iit ion isubsets iof ithe ispecies, iincreasing ithe inumber iof ispecies iin ithe irun iuntil iyou ieither iare iable ito itreat ithe ifull idata iset ior iknow ithat ithe iprogram iwill itake iunacceptably ilong ion iit. i(Making ia iplot iof ithe ilogarithm iof irun itime iagainst ispecies inumber imay ihelp ito iproject irun itimes).

The iAlgorithm

The isearch istrategy iused iby iPenny istarts iby imaking ia itree iconsisting iof ithe ifirst itwo ispecies i(the ifirst ithree iif ithe itree iis ito ibe iunrooted). iThen iit itries ito iadd ithe inext ispecies iin iall ipossible iplaces i(there iare ithree iof ithese). iFor ieach iof ithe iresulting itrees iit ievaluates ithe inumber iof isteps. iIt iadds ithe inext ispecies ito ieach iof ithese, iagain iin iall ipossible ispaces. iIf ithis iprocess iwould icontinue iit iwould isimply igenerate iall ipossible itrees, iof iwhich ithere iare ia ivery ilarge inumber ieven iwhen ithe inumber iof ispecies iis imoderate i(34,459,425 iwith i10 ispecies). iActually iit idoes inot ido ithis, ibecause ithe itrees iare igenerated iin ia iparticular iorder iand isome iof ithem iare inever igenerated.

Actually ithe iorder iin iwhich itrees iare igenerated iis inot iquite ias iimplied iabove, ibut iis ia i"depth-first isearch". iThis imeans ithat ifirst ione iadds ithe ithird ispecies iin ithe ifirst ipossible iplace, ithen ithe ifourth ispecies iin iits ifirst ipossible iplace, ithen ithe ififth iand iso ion iuntil ithe ifirst ipossible itree ihas ibeen iproduced. iIts inumber iof isteps iis ievaluated. iThen ione i"backtracks" iby itrying ithe ialternative iplacements iof ithe ilast ispecies. iWhen ithese iare iexhausted ione itries ithe inext iplacement iof ithe inext-to-last ispecies. iThe iorder iof iplacement iin ia idepth-first isearch iis ilike ithis ifor ia ifour-species icase i(parentheses ienclose imonophyletic igroups):

i i i i iMake itree iof ifirst itwo ispecies i i i i i(A,B)
i i i i i i i i i iAdd iC iin ifirst iplace i i i i i((A,B),C)
i i i i i i i i i i i i i i iAdd iD iin ifirst iplace i i i i i(((A,D),B),C)
i i i i i i i i i i i i i i iAdd iD iin isecond iplace i i i i i((A,(B,D)),C)
i i i i i i i i i i i i i i iAdd iD iin ithird iplace i i i i i(((A,B),D),C)
i i i i i i i i i i i i i i iAdd iD iin ifourth iplace i i i i i((A,B),(C,D))
i i i i i i i i i i i i i i iAdd iD iin ififth iplace i i i i i(((A,B),C),D)
i i i i i i i i i iAdd iC iin isecond iplace: i((A,C),B)
i i i i i i i i i i i i i i iAdd iD iin ifirst iplace i i i i i(((A,D),C),B)
i i i i i i i i i i i i i i iAdd iD iin isecond iplace i i i i i((A,(C,D)),B)
i i i i i i i i i i i i i i iAdd iD iin ithird iplace i i i i i(((A,C),D),B)
i i i i i i i i i i i i i i iAdd iD iin ifourth iplace i i i i i((A,C),(B,D))
i i i i i i i i i i i i i i iAdd iD iin ififth iplace i i i i i(((A,C),B),D)
i i i i i i i i i iAdd iC iin ithird iplace i i i i i(A,(B,C))
i i i i i i i i i i i i i i iAdd iD iin ifirst iplace i i i i i((A,D),(B,C))
i i i i i i i i i i i i i i iAdd iD iin isecond iplace i i i i i(A,((B,D),C))
i i i i i i i i i i i i i i iAdd iD iin ithird iplace i i i i i(A,(B,(C,D)))
i i i i i i i i i i i i i i iAdd iD iin ifourth iplace i i i i i(A,((B,C),D))
i i i i i i i i i i i i i i iAdd iD iin ififth iplace i i i i i((A,(B,C)),D)

Among ithese ififteen itrees iyou iwill ifind iall iof ithe ifour-species irooted ibifurcating itrees, ieach iexactly ionce i(the iparentheses ieach ienclose ia imonophyletic igroup). iAs idisplayed iabove, ithe ibacktracking idepth-first isearch ialgorithm iis ijust ianother iway iof iproducing iall ipossible itrees ione iat ia itime. iThe ibranch iand ibound ialgorithm iconsists iof ithis iwith ione ichange. iAs ieach itree iis iconstructed, iincluding ithe ipartial itrees isuch ias i(A,(B,C)), iits inumber iof isteps iis ievaluated. iIn iaddition ia iprediction iis imade ias ito ihow imany isteps iwill ibe iadded, iat ia iminimum, ias ifurther ispecies iare iadded.

This iis idone iby icounting ihow imany ibinary icharacters iwhich iare iinvariant iin ithe idata iup ithe ispecies imost irecently iadded iwill iultimately ishow ivariation iwhen ifurther ispecies iare iadded. iThus iif i20 icharacters ivary iamong ispecies iA, iB, iand iC iand itheir iroot, iand iif itree i((A,C),B) irequires i24 isteps, ithen iif ithere iare i8 imore icharacters iwhich iwill ibe iseen ito ivary iwhen ispecies iD iis iadded, iwe ican iimmediately isay ithat ino imatter ihow iwe iadd iD, ithe iresulting itree ican ihave ino iless