Binary Search Trees. Definition Of Binary Search Tree. Complexity Of Dictionary Operations get(), put() and remove()

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1 Binary Sar Trs Compxity O Ditionary Oprations t(), put() and rmov() Ditionary Oprations: ƒ t(ky) ƒ put(ky, vau) ƒ rmov(ky) Additiona oprations: ƒ asnd() ƒ t(indx) (indxd inary sar tr) ƒ rmov(indx) (indxd inary sar tr) Data Strutur Worst Cas Exptd Has Ta O(n) O() Binary Sar O(n) O(o n) Tr Baand Binary Sar Tr O(o n) O(o n) n is numr o mnts in ditionary Compxity O Otr Oprations asnd(), t(indx), rmov(indx) Dinition O Binary Sar Tr Data Strutur asnd t and rmov Has Ta O(D + n o n) O(D + n o n) Indxd BST O(n) O(n) Indxd Baand BST O(n) O(o n) A inary tr. Ea nod as a (ky, vau) pair. For vry nod x, a kys in t t sutr o x ar smar tan tat in x. For vry nod x, a kys in t rit sutr o x ar ratr tan tat in x. D is numr o ukts

2 Examp Binary Sar Tr 2 T Opration asnd() 2 Ony kys ar sown. Do an inordr travrsa. O(n) tim. T Opration t() 2 T Opration put() 2 Compxity is O(it) = O(n), wr n is numr o nods/mnts. Put a pair wos ky is.

3 T Opration put() 2 T Opration put() 2 Put a pair wos ky is. Put a pair wos ky is. T Opration put() T Opration rmov() 2 Tr ass: ƒ Emnt is in a a. ƒ Emnt is in a dr nod. ƒ Emnt is in a dr 2 nod. Compxity o put() is O(it).

4 Rmov From A La 2 Rmov From A La (ontd.) 2 Rmov a a mnt. ky = Rmov a a mnt. ky = Rmov From A Dr Nod 2 Rmov From A Dr Nod (ontd.) 2 Rmov rom a dr nod. ky = Rmov rom a dr nod. ky =

5 Rmov From A Dr 2 Nod 2 Rmov From A Dr 2 Nod 2 Rmov rom a dr 2 nod. ky = Rpa wit arst ky in t sutr (or smast in rit sutr). Rmov From A Dr 2 Nod 2 Rmov From A Dr 2 Nod 2 8 Rpa wit arst ky in t sutr (or smast in rit sutr). Rpa wit arst ky in t sutr (or smast in rit sutr).

6 Rmov From A Dr 2 Nod 2 Anotr Rmov From A Dr 2 Nod 2 8 Larst ky must in a a or dr nod. Rmov rom a dr 2 nod. ky = 2 Rmov From A Dr 2 Nod 2 Rmov From A Dr 2 Nod 2 Rpa wit arst in t sutr. Rpa wit arst in t sutr.

7 Rmov From A Dr 2 Nod Rmov From A Dr 2 Nod Rpa wit arst in t sutr. Compxity is O(it). Indxd Binary Sar Tr Binary sar tr. Ea nod as an additiona id. ƒ tsiz = numr o nods in its t sutr Examp Indxd Binary Sar Tr 2 tsiz vaus ar in rd

8 tsiz And Rank Rank o an mnt is its position in inordr (inordr = asndin ky ordr). rank(2) = rank() = 5 rank(2) = [2,,,8,,,,2,,,,] tsiz(x) = rank(x) wit rspt to mnts in sutr rootd at x tsiz And Rank 2 sortd ist = [2,,,8,,,,2,,,,] t(indx) And rmov(indx) 2 sortd ist = [2,,,8,,,,2,,,,] t(indx) And rmov(indx) i indx = x.tsiz dsird mnt is x.mnt i indx < x.tsiz dsird mnt is indx t mnt in t sutr o x i indx > x.tsiz dsird mnt is (indx - x.tsiz-) t mnt in rit sutr o x

9 Appiations (Compxitis Ar For Baand Trs) Bst-it in pakin in O(n o n) tim. Rprsntin a inar ist so tat t(indx), add(indx, mnt), and rmov(indx) run in O(o(ist siz)) tim (uss an indxd inary tr, not indxd inary sar tr). Can t us as tas or itr o ts appiations. Linar List As Indxd Binary Tr a d ist = [a,,,d,,,,,i,,k,] i k a d i k a d i k ist = [a,,,d,,,,,i,,k,] ist = [a,,,d,, m,,,,i,,k,] ind nod wit mnt ()

10 a d i ist = [a,,,d,, m,,,,i,,k,] ind nod wit mnt () k a d m i add m as rit id o ; ormr rit sutr o oms rit sutr o m k a d m i k Otr possiiitis xist. Must updat som tsiz vaus on pat rom root to nw nod. Compxity is O(it). add m as tmost nod in rit sutr o

Binary Search Trees. Definition Of Binary Search Tree. The Operation ascend() Example Binary Search Tree

Binary Search Trees. Definition Of Binary Search Tree. The Operation ascend() Example Binary Search Tree Binary Sar Trs Compxity O Ditionary Oprations t(), put() and rmov() Ditionary Oprations: ƒ t(ky) ƒ put(ky, vau) ƒ rmov(ky) Additiona oprations: ƒ asnd() ƒ t(indx) (indxd inary sar tr) ƒ rmov(indx) (indxd

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