Give asymptotic upper and lower bounds for T(n) in each of the 给T渐近上界和下界（N）的每个

Give asymptotic upper and lower bounds for T(n) in each of the 给T渐近上界和下界（N）的每个 ?????? ???? ?????? , ???? ????? ?????? ????? ????????? ??? ?? ?? )????????? ???( ????? ??? ??? ., .n?2 ???? T(n) = O(1) ?? ???a) T(n) = T(n – 1) + n b) T(n) = T(n – 1) + logn 9n,,T(n),T，n c) ,,10,, n,,2T(n),16T，nd) ,,4,, nnT(n),3T(，5)，e) 32 1T(n),T(n,1)，f) n nT(n),9T()，ng) 3 ?????? ?????????? ??? ?? ?? )????????? ???( ????? ??? ??? .2 a. int exp(int x, int n) { if (n == 0) return 1; else if (n == 1) return x; else return x*exp(x, n-1); } b. k = 0; for(int i = n; i >= n; i--) k += exp(x, i); Note: ‘exp’ is a function from the previous example. c. void func2( int n ) { if ( n<=0 ) return 1; else if (n % 2 ) return func2( n/2 ); else 1 ); return func2( n – } .3 T(n) = 2T(n/2) + 6n – 1 n ? 2 ???? T(1) = 1 ???? .? 2 T(n) = O(n) :???? ?? ???? ???? .?2f ( n ) = f ( n / 2 ) + 2 log n f ( 1 ) = f ( 0 ) = 1 :???? ?? ????3 f ( n ) = O ( log n ) .4 ?? ???-? ?????? ?????? ????? ????? .??????? ????? ?????? n ?? ???? ???? AA . ???? ????? ???????? ??? , AO(nloglogn)logn .5 ?? ???? .???? ,1-? ?????? ????????? ??? ?? ,??????? n ???? ?????? ?????? ???? .O(n) ??????? ?? ???? ??? .?-? ???? ??????? ??????? – ??????? :????? ??????? ??? ?????? ?? ????????? ????? ????? ???? T ?????? ?? .6 ??? ?? ??? T .? :??? ????????? ????? ??????? TR ???? ???? TL ????? ??? .? | NumberOfNodes(TL) - NumberOfNodes(TR) | ? 1 .O(log n) ??? ???????? n ?? T ?? ?? ???? ?? ???? )????? ?????( ?????? ????? ?? ?? preorder ???? ????? .7 20, 15, 10, 5, 2, 8, 13, 18, 30, 40, 35, 37 ?????? ?????? ????? ?? ?? ???? ??-?? ??? ????? ??? ?? true ??????? ???????? ????? ??? ??? ???? ?????? ?????? ???? .8 .??????? ?? ???? ??? ???? .???? ?? ???? ??? ???? .?????? ????? ?? ??? ?? ?? true ??????? ?????? ?? ???? ?????? ?????? ???? .9 .??????? ?? ??? .????? ???? ???? ????? ??????? ????? ???? ???? ?? ?? ?????? ?????? ???? ??? .,1 .????? ??? ?????? ???? ???? ??????? ?????? ??????????? ?? .O(n logn)- ????? ?? ???? ,?????? ?????? n ?? ?????? ?? ????? ?????? – Init . O(log n) ????? ???? ??? .????? x ?? ???? – Insert(x) . O(1) ????? ???????? / ???????? ???? ?? ???? – Find_min, Find_max . O(log n) ????? ???? ??? .?????? ???? ?? ???? – Find_med . O(log n) ?????? ???????? / ???????? ???? ?? ??? – Del_min, Del_max . O(log n) ????? ???? ??? .?????? ?????? ???? ?? ??? – Del_med .????? ?? ???? ?????? ???? ???? ?? ??? .????? ???? ???? ??? ,5 ,,1 ,35 ?????? ???? .,, ???? 6, 71 23 ???? ???? 2 ???? ???? 95 52 5 ????????? ?? ????? 2-3 ??? .,2Tour(x) if x is a leaf then print (x->value); return; if x has two children Tour(right child); Tour(right child); Tour(left child); if x has three children Tour(leftmost child); Tour(rightmost child) ??? ?? ?? ?????? ???? ???? ????? n ??? 2-3 ??? Tour(T) ?? T(n) ???? ??? ??)? .??????? ?? ?????? ????? ????? ???? ?? ?? ?????? ???? ???? ????? n ??? 2-3 ??? Tour(T) ?? H(n) ???? ??? ?? )? .??????? ?? ?????? .?????? ??? ????? )? ,T(n),,(H(n)) T(n),O(H(n)) , :??? ?? ?? ???? Tour(T) ?? ???? ???? ?? )? 16,- 14,- 20,- 17,- 10,13 15,- 30,40 20 5 10 13 14 15 16 17 30 40 :????? ??????? ?? ????? ?? ?????? .2-3 ??? ???? ???? ?? – ????? ???? ??? ?????? .3,insert( 1 ), insert( 2 ), …, insert( 10 ) (?????? ,1 ?"??) .?????? ,1 ???? ?????? ??? ?? ?? ?????.? .)21 - ?( ???? ?????? ???? ???? 1,2,…,n ?????? ?? ????? ?????? ?? ???? ????? ??? .?????? ???????? ????? ???? ?? ?? ????? ????? ???? ?? ?? ????? .? ????? ???? ??? ?? ???? 3 ,???? 2( 2-3 ?? ?? ???????? ?????? ???? ?? ???? ??? .? .?????? ?)??????? .?????? ???????? ???? ???? ???? ??? ?????? ?????? ??????? ???? .?