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AB-treeTis a rooted tree (whose root is root[T]) having the following properties:

1. Every node xhas the following fields:

a. n[x], the number of eys !urrently stored in node x,

b. the n[x] eys themselves, stored in non de!reasing order, so that key"[x]

# key1[x] # $$$ # keyn[x]%1[x],

!. leaf [x], a boolean value that is &'E if xis a leaf and A*+E if xis

an internal node.

. Ea!h internal nodexalso !ontains n[x]- 1 pointers c"[x], c1[x], ..., cn[x][x] to its

!hildren. *eaf nodes have no !hildren, so their cifields are undefined.

. &he eys keyi[x] separate the ranges of eys stored in ea!h subtree: if kiis any

ey stored in the subtree with root ci[x], then

k"# key"[x] # k1# key1[x] #$$$ # keyn[x]%1[x] # kn[x].

/. All leaves have the same depth, whi!h is the tree0s height h.

. &here are lower and upper bounds on the number of eys a node !an !ontain.&hese bounds !an be e2pressed in terms of a fi2ed integer t3 !alled the

minimum degreeof the 4%tree:

a. Every node other than the root must have at least t% 1 eys. Every

internal node other than the root thus has at least t!hildren. 5f the tree

is nonempty, the root must have at least one ey.

b. Every node !an !ontain at most t% 1 eys. &herefore, an internal node

!an have at most t!hildren. 6e say that a node isfullif it !ontains

e2a!tly t% 1 eys.

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4%&ree%+ear!h(2, )

i keyi[x] do i

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ow before inserting, the root is splinted:

inally after the last slinting:

8 =,,6

&

;

8 = *,@, ',+ >,6

, < &

;

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. +how all legal 4%trees of minimum degree that represent 1, , , /, B.

solution:

we now that every node % e2!ept the root % must have at list t%1 eys, that

and at most t%1 eys. Also remember that the leaves stay in the same depth:

1.

.

.

/.

. 6hy don0t we allow a minimum degree of tC 1D

Solution:

6e get a 4%tree nodes with no eys.

1 ,/,

1, /,

/

1,,

/

1

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/. E2plain how to find the ma2imum ey stored in a 4%tree and how to find the

prede!essor of a given ey stored in a 4%tree.

Solution:

inding the ma2imum in a 4%tree is ?uite similar to finding a ma2imum in a binary

sear!h tree we need to find the right most leaf for the given root, and return the last

ey.

ind%@a2imum(2) FF assumption: the tree is not emptyif leaf[2]

retrun (eyn[2]%1[2])

else

return ind%@a2imum(8n[2][2])

8omple2ity: in the height of tree 9(logtn).

inding the prede!essor of eyi[2] (given 2 and i ) is a!!ording to the following rools:

1. if the 2 is not a leaf return the ma2imum of 8i[2] (every ey in this subtree

apply keyi%1[x] # k# keyi[x] ,if keyi%1[x] e2ist)

. otherwise (it is a leaf) if iG" we return eyi%1[2].

if the above !onditions are not satisfied, we loo for the first prede!essor node

y, that its pointer 8i[y] is also the prede!essor of 2 and iG" we return ey i%1[y], if su!h node y is not e2ist, nil is returned.

ote: we add another field p for ea!h node that points to it0s father.

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Hrede!essor(2,i ) FF we want the prede!essor to of ey i[2]if not leaf[2]

return ind%@a2imum(8i[2])

else if iG"

return eyi%1[2]

else

found

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