Theorem 1. An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.

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Hilary Lloyd
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1 Cptr 11: Trs Introuton to Trs Dnton 1 (Tr). A tr s onnt unrt rp wt no sp ruts. Tor 1. An unrt rp s tr n ony tr s unqu sp pt twn ny two o ts vrts. Dnton 2. A root tr s tr n w on vrtx s n snt s t root n vry s rt wy ro t root. T onvnton s to rw root trs wt t root t t top, n tn ot t rton rrows on t s. Dnton 3. Lt T root tr. I v s vrtx n T otr tn t root, tn t prnt o v s t unqu vrtx u su tt tr s rt ro u to v. Wn u s prnt o v, v s o u. Vrts wt t s prnt r sns. T nstors o vrtx otr tn t root r t vrts n t pt ro t root to ts vrtx, xun t vrtx ts ut nun t root. T snnts o vrtx v r tos vrts tt v v s n nstor. A vrtx o root tr s t s no rn. Vrts tt v rn r ntrn vrts. T root s n ntrn vrtx unss t s t ony vrtx (n w s t s ). I s vrtx n tr, t sutr wt s ts root s t surp o t tr onsstn o, ts snnts, n s nnt to tos snnts. Exp 1. 1

2 Dnton 4. An orr tr s tr n w n orrn s ssn to t rn o vry ntrn vrtx. Wn rwn t tr, w orr t rn t to rt. Wn tr r t ost two rn o ny vn vrtx, w us t n rt to sr t (otrws w us rst, son, t.). Exp 2. Dnton 5 (-ry Tr). A root tr s n -ry tr vry ntrn vrtx s no or tn rn. An -ry tr s u vry ntrn vrtx s xty rn. An -ry tr wt = 2 s nry tr Tor 2. A tr wt n vrts s n 1 s. Tor 3. A u -ry tr wt () n vrts s = n 1 ntrn vrts n = n( 1) + 1 () ntrn vrts s n = + 1 vrts n = ( 1) + 1. () vs s n = 1 1 vrts n = 1 1 ntrn vrts. vs. (So, vn n u -ry tr n on o n,, or, w n n t otr two quntts.) 2

3 Exp 3. Suppos tt soon strts n. E prson wo rvs t s s to orwr t to our otr pop. So pop o ts, ut otrs o not sn ny s. How ny pop v sn t, nun t rst prson, no on rvs t or tn on n t n ns tr tr v n 100 pop wo r t ut not sn t out? How ny pop snt out t? Dnton 6. T v o vrtx s ts stn ro t root. T root s t v 0, ts rn r t v 1, t. T t o tr s t xu o t vs o ts vrts. A tr o t s s to n o ts vs r t t or 1. Exp 4. Tor 4. Tr r t ost vs n n -ry tr o t. Corory 5. I n -ry tr o t s vs, tn o. I t -ry tr s u n n, tn = o. 3

4 Apptons o Trs Exp 5 (Bnry Sr Tr). Crt nry sr tr or ts st o ns: Crott, Ros, Arturo, Kr, Constn, Knnt, Tr, Jss, Fr, M. Exp 6 (Dson Tr). Crt son tr tt orrs t nts,,. Exp 7 (G Trs). Crt tr or N. 4

5 Exp 8 (Hun Con). W on t two trs (ou ust sn vrtx) wt st tot wt nto nw tr wt t rr wt on t t n t sr wt on t rt. On ts pross s opt w r o t o ro root to. Crtr Proty o Ourrn o 0.14 p 0.03 r 0.11 s 0.20 t Tr Trvrs Aort 6 (Unvrs Arss Syst). 1. L t root wt t ntr 0. Tn ts rn (t v 1) ro t to rt wt 1, 2, 3,...,. 2. For vrtx v t v n wt A, ts v rn, s ty r rwn, ro t to rt, wt A.1, A.2, A.3,..., A. v. Exp 9. 5

6 Aort 7 (Prorr Trvrs). Lt T n orr tr wt root r. I T ontns ony r, t r s t prorr trvrs o T. Otrws, suppos tt T 1, T 2,..., T n r t sutrs t r ro t to rt n T. T prorr trvrs ns t r. It ontnus y trvrsn T 1 n prorr, tn T 2 n prorr, n so on, unt T n s trvrs n prorr. Not: Prorr trvrs vs t s orrn s unvrs rss syst. Exp 10. 6

7 Aort 8 (Inorr Trvrs). Lt T n orr tr wt root r. I T ontns ony r, t r s t norr trvrs o T. Otrws, suppos tt T 1, T 2,..., T n r t sutrs t r ro t to rt n T. T norr trvrs ns y trvrsn T 1 n norr, tn vstn r. It ontnus y trvrsn T 2 n norr, tn T 3 n norr, n so on, unt T n s trvrs n norr. Exp 11. 7

8 Aort 9 (Postorr Trvrs). Lt T n orr tr wt root r. I T ontns ony r, t r s t postorr trvrs o T. Otrws, suppos tt T 1, T 2,..., T n r t sutrs t r ro t to rt n T. T postorr trvrs ns y trvrsn T 1 n postorr, tn T 2 n postorr, n so on, unt T n s trvrs n postorr, n ns y vstn r. Exp 12. 8

9 An sr wy to n t trvrss. Drw urv roun t s o t tr, strtn t t root n ovn own t t rn, n nn t t root. To n t prorr trvrs, st vrtx t rst t ts urv psss t. To n t norr trvrs, st t rst t ts urv psss t n n ntrn vrtx t son t ts urv psss t. To n t postorr trvrs, st vrtx t st t ts urv psss t on t wy up to ts prnt. Exp 13. 9

10 11.4 n Mnu Spnnn Trs Dnton 7. Lt G sp rp. A spnnn tr o G s surp o G tt s tr ontnn vry vrtx o G. Tor 10. A sp rp s onnt n ony t s spnnn tr. Exp 14. Dnton 8. A nu spnnn tr n onnt wt rp s spnnn tr tt s t sst poss su o wts o ts s. 10

Applications of trees

Applications of trees Trs Apptons o trs Orgnzton rts Attk trs to syst Anyss o tr ntworks Prsng xprssons Trs (rtrv o norton) Don-n strutur Mutstng Dstnton-s orwrng Trnsprnt swts Forwrng ts o prxs t routrs Struturs or nt pntton