BinaryTrees

S.M.NiazArifin,PhDComputerScienceandEngineering

UniversityofNotreDameIndiana,USA

•  Binarytree•  defini=on,example

•  types

•  Binarysearchtree•  defini=on,example

•  opera=ons– search– maximum,minimum–  insert–  traversal– delete

S.M.NiazArifin,PhD

Overview

•  atreedatastructure•  eachnodehasatmosttwochildren–  thele)childandtherightchild– mayalsocontainaparentnode

•  leafnodes:lowest-levelnodes– nochildren

•  alwaysrooted•  root:onlynodewhoseparentisNIL

S.M.NiazArifin,PhD

BinaryTree

S.M.NiazArifin,PhD

BinaryTree:Example

root

internalnodes

leafnodes

•  numberofnodes,n=9•  height(depth),h=3

•  fullbinarytree•  a.k.a.aproperorplanebinarytree

•  everynodeinthetreehaseither0or2children

S.M.NiazArifin,PhD

BinaryTree:Types

•  completebinarytree•  everylevel,exceptpossiblythelast,iscompletelyfilled

•  allnodesinthelastlevelareasfarle)aspossible

S.M.NiazArifin,PhD

BinaryTree:Types

•  balancedbinarytree•  hastheminimumpossibleheight(a.k.a.depth)fortheleafnodes

•  self-balancing– automa=callykeepsitsheightsmall

S.M.NiazArifin,PhD

BinaryTree:Types

balancedbinarytree

S.M.NiazArifin,PhD

BinaryTree:Types

unbalancedbinarytree

imagesource:wikipedia

•  ordered(sorted)binarytree•  nodekeysarealwaysstoredinsuchawayastosa=sfytheBSTproperty:–  letxbeanodeinabinarysearchtree–  ifyisanodeintheleRsubtreeofx,theny:key≤x:key–  ifyisanodeintherightsubtreeofx,theny:key≥x:key

S.M.NiazArifin,PhD

BinarySearchTree(BST)

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

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31 44 9066

root

ifyisanodeintheleRsubtreeofx,theny:key≤x:key

ifyisanodeintherightsubtreeofx,theny:key≥x:key

S.M.NiazArifin,PhD

BinarySearchTree(BST)

•  allowsfastsearch(lookup),inser=on,anddele=onofnodes

•  supportsmanydynamic-setopera=ons– create,add,remove,etc.

•  let,n=numberofnodesinthetree

•  'mecomplexityforsearch,inser=on,ordele=on– propor=onaltothelogarithmofn:O(logn)

•  averagecase[worstcase:O(n)]– muchbeWerthanlinear=meforunsortedarrays:O(n)

S.M.NiazArifin,PhD

BinarySearchTree:Proper=es

•  search•  maximum,minimum

•  insert•  traversal•  delete

S.M.NiazArifin,PhD

BinarySearchTree:Opera=ons

•  searchthetreeforaspecifickey•  canberecursiveoritera;ve•  example:searchfor45

S.M.NiazArifin,PhD

Search

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

1.  startattheroot,45>25–  searchinrightsubtree

S.M.NiazArifin,PhD

Search

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

1.  startattheroot,45>25–  searchinrightsubtree

2.   45<50,searchin50’sleRsubtree

S.M.NiazArifin,PhD

Search

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

1.  startattheroot,45>25–  searchinrightsubtree

2.   45<50,searchin50’sleRsubtree3.   45>35,searchin35’srightsubtree

S.M.NiazArifin,PhD

Search

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

1.  startattheroot,45>25–  searchinrightsubtree

2.   45<50,searchin50’sleRsubtree3.   45>35,searchin35’srightsubtree4.   45>44,but44hasnorightsubtree

so45is�notintheBST

S.M.NiazArifin,PhD

Search

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  algorithm•  n=numberofnodesintree

•  ;mecomplexity:O(logn)

S.M.NiazArifin,PhD

Search

290 Chapter 12 Binary Search Trees

2 4

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Figure 12.2 Queries on a binary search tree. To search for the key 13 in the tree, we follow the path15 ! 6 ! 7 ! 13 from the root. The minimum key in the tree is 2, which is found by followingleft pointers from the root. The maximum key 20 is found by following right pointers from the root.The successor of the node with key 15 is the node with key 17, since it is the minimum key in theright subtree of 15. The node with key 13 has no right subtree, and thus its successor is its lowestancestor whose left child is also an ancestor. In this case, the node with key 15 is its successor.

TREE-SEARCH.x; k/

1 if x == NIL or k == x:key2 return x3 if k < x:key4 return TREE-SEARCH.x: left; k/5 else return TREE-SEARCH.x:right; k/

The procedure begins its search at the root and traces a simple path downward inthe tree, as shown in Figure 12.2. For each node x it encounters, it compares thekey k with x:key. If the two keys are equal, the search terminates. If k is smallerthan x:key, the search continues in the left subtree of x, since the binary-search-tree property implies that k could not be stored in the right subtree. Symmetrically,if k is larger than x:key, the search continues in the right subtree. The nodesencountered during the recursion form a simple path downward from the root ofthe tree, and thus the running time of TREE-SEARCH is O.h/, where h is the heightof the tree.

We can rewrite this procedure in an iterative fashion by “unrolling” the recursioninto a while loop. On most computers, the iterative version is more efficient.

12.2 Querying a binary search tree 291

ITERATIVE-TREE-SEARCH.x; k/

1 while x ¤ NIL and k ¤ x:key2 if k < x:key3 x D x: left4 else x D x:right5 return x

Minimum and maximumWe can always find an element in a binary search tree whose key is a minimum byfollowing left child pointers from the root until we encounter a NIL, as shown inFigure 12.2. The following procedure returns a pointer to the minimum element inthe subtree rooted at a given node x, which we assume to be non-NIL:

TREE-MINIMUM.x/

1 while x: left ¤ NIL2 x D x: left3 return x

The binary-search-tree property guarantees that TREE-MINIMUM is correct. If anode x has no left subtree, then since every key in the right subtree of x is at least aslarge as x:key, the minimum key in the subtree rooted at x is x:key. If node x hasa left subtree, then since no key in the right subtree is smaller than x:key and everykey in the left subtree is not larger than x:key, the minimum key in the subtreerooted at x resides in the subtree rooted at x: left.

The pseudocode for TREE-MAXIMUM is symmetric:

TREE-MAXIMUM.x/

1 while x:right ¤ NIL2 x D x:right3 return x

Both of these procedures run in O.h/ time on a tree of height h since, as in TREE-SEARCH, the sequence of nodes encountered forms a simple path downward fromthe root.

Successor and predecessorGiven a node in a binary search tree, sometimes we need to find its successor inthe sorted order determined by an inorder tree walk. If all keys are distinct, the

•  maximum

•  minimum

•  ;mecomplexity:O(logn)

S.M.NiazArifin,PhD

MaximumandMinimum

12.2 Querying a binary search tree 291

ITERATIVE-TREE-SEARCH.x; k/

1 while x ¤ NIL and k ¤ x:key2 if k < x:key3 x D x: left4 else x D x:right5 return x

Minimum and maximumWe can always find an element in a binary search tree whose key is a minimum byfollowing left child pointers from the root until we encounter a NIL, as shown inFigure 12.2. The following procedure returns a pointer to the minimum element inthe subtree rooted at a given node x, which we assume to be non-NIL:

TREE-MINIMUM.x/

1 while x: left ¤ NIL2 x D x: left3 return x

The binary-search-tree property guarantees that TREE-MINIMUM is correct. If anode x has no left subtree, then since every key in the right subtree of x is at least aslarge as x:key, the minimum key in the subtree rooted at x is x:key. If node x hasa left subtree, then since no key in the right subtree is smaller than x:key and everykey in the left subtree is not larger than x:key, the minimum key in the subtreerooted at x resides in the subtree rooted at x: left.

The pseudocode for TREE-MAXIMUM is symmetric:

TREE-MAXIMUM.x/

1 while x:right ¤ NIL2 x D x:right3 return x

Both of these procedures run in O.h/ time on a tree of height h since, as in TREE-SEARCH, the sequence of nodes encountered forms a simple path downward fromthe root.

Successor and predecessorGiven a node in a binary search tree, sometimes we need to find its successor inthe sorted order determined by an inorder tree walk. If all keys are distinct, the

12.2 Querying a binary search tree 291

ITERATIVE-TREE-SEARCH.x; k/

1 while x ¤ NIL and k ¤ x:key2 if k < x:key3 x D x: left4 else x D x:right5 return x

Minimum and maximumWe can always find an element in a binary search tree whose key is a minimum byfollowing left child pointers from the root until we encounter a NIL, as shown inFigure 12.2. The following procedure returns a pointer to the minimum element inthe subtree rooted at a given node x, which we assume to be non-NIL:

TREE-MINIMUM.x/

1 while x: left ¤ NIL2 x D x: left3 return x

The binary-search-tree property guarantees that TREE-MINIMUM is correct. If anode x has no left subtree, then since every key in the right subtree of x is at least aslarge as x:key, the minimum key in the subtree rooted at x is x:key. If node x hasa left subtree, then since no key in the right subtree is smaller than x:key and everykey in the left subtree is not larger than x:key, the minimum key in the subtreerooted at x resides in the subtree rooted at x: left.

The pseudocode for TREE-MAXIMUM is symmetric:

TREE-MAXIMUM.x/

1 while x:right ¤ NIL2 x D x:right3 return x

Both of these procedures run in O.h/ time on a tree of height h since, as in TREE-SEARCH, the sequence of nodes encountered forms a simple path downward fromthe root.

Successor and predecessorGiven a node in a binary search tree, sometimes we need to find its successor inthe sorted order determined by an inorder tree walk. If all keys are distinct, the

•  insertnewnodex:worksmuchlikesearch•  ;mecomplexity:O(lgn)•  setrootascurrentnode(current)•  recursivelyexaminecurrent•  ifx:key<current:key–  ifcurrenthasaleRchild,searchleR– elseaddxascurrent’sleRchild

•  ifx:key≥current:key–  ifcurrentnodehasarightchild,searchright– elseaddxascurrent’srightchild

S.M.NiazArifin,PhD

Insert

•  example:insertnode60

S.M.NiazArifin,PhD

Insert

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

1.  startattheroot,60>25–  searchinrightsubtree

S.M.NiazArifin,PhD

Insert

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

1.  startattheroot,60>25–  searchinrightsubtree

2.   60>50,searchin50’srightsubtree

S.M.NiazArifin,PhD

Insert

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

1.  startattheroot,60>25–  searchinrightsubtree

2.   60>50,searchin50’srightsubtree3.   60<70,searchin70’sleRsubtree

S.M.NiazArifin,PhD

Insert

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

CS21, Tia Newhall

25

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root (1)

(2)(3)

(4)

Example: insert 60 in the tree:1. start at the root, 60 is greater than 25, search in right subtree2. 60 is greater than 50, search in 50’s right subtree3. 60 is less than 70, search in 70’s left subtree4. 60 is less than 66, add 60 as 66’s left child

60

1.  startattheroot,60>25–  searchinrightsubtree

2.   60>50,searchin50’srightsubtree3.   60<70,searchin70’sleRsubtree4.   60<66,add60as66’sleRchild

S.M.NiazArifin,PhD

Insert

•  visiteverynodeinthetree•  threestepstoatraversal:

1.  visitthecurrentnode2.  traverseitsle)subtree3.  traverseitsrightsubtree

•  orderdefinesdifferenttraversalmethods:– pre-ordertraversal:(1)(2)(3)–  in-ordertraversal:(2)(1)(3)– post-ordertraversal:(2)(3)(1)

•  ;mecomplexity:O(logn)

S.M.NiazArifin,PhD

Traversal

•  example:in-ordertraversal:(2)(1)(3)

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,10,

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,10,12,

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,10,12,15,

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,10,12,15,

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,10,12,15,18,

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,10,12,15,18,22,

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,10,12,15,18,22,24,•  andsoon…

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

•  example:in-ordertraversal:(2)(1)(3)

•  4,10,12,15,18,22,24,…

S.M.NiazArifin,PhD

Traversal

CS21, Tia Newhall

Binary Search Trees (BST)1. Hierarchical data structure with a single pointer to root node 2. Each node has at most two child nodes (a left and

a right child)3. Nodes are organized by the Binary Search property:

• Every node is ordered by some key data field(s)• For every node in the tree, its key is greater than its

left child’s key and less than its right child’s key

25

15

10 22

4 12 2418

50

35 70

31 44 9066

root

sortedorder

•  threepossiblecases:1.  dele=nganodewithnochildren2.  dele=nganodewithonechild3.  dele=nganodewithtwochildren

•  ;mecomplexity:O(logn)

S.M.NiazArifin,PhD

Delete

1.  dele=nganodewithnochildren–  simplyremovethenodefromthetree

S.M.NiazArifin,PhD

Delete

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deletenode7

2. dele=nganodewithonechild–  removethenodeandreplaceitwithitschild

S.M.NiazArifin,PhD

Delete

deletenode1

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3. dele=nganodewithtwochildren•  hardercase

•  let,nodez(havingtwochildren)tobedeleted

•  iden=fyz’ssuccessor,y

•  successorofanodez–  thenodeywiththesmallestkeyintree

suchthaty:key>z:key

S.M.NiazArifin,PhD

Delete

3.  dele=nganodewithtwochildren•  z’ssuccessor,y

•  yeitherisaleaforhasonlytherightchild1.  promoteytoz’splace2.  treatthelossofyusingoneoftheabovetwo

solu=ons

S.M.NiazArifin,PhD

Delete

3.  dele=nganodewithtwochildren•  todelete:z=node8;

•  successor,y=node9–  smallestkeyintreewithy:key>z:key

S.M.NiazArifin,PhD

Delete

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3.  dele=nganodewithtwochildren•  todelete:z=node8;•  successor,y=node9

1.  promotenode9tonode8’splace2.  treatthelossofnode9using:•  dele=nganodewithonechild

S.M.NiazArifin,PhD

Delete

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•  Binarytree

•  Binarysearchtree– defini=on,example– opera=ons

•  search•  maximum,minimum

•  insert•  traversal•  delete

S.M.NiazArifin,PhD

Summary

•  Red-blacktree•  aspecialtypeofself-balancingBST•  eachnodehasanextracolorbit–  redorblack

•  colorbitsareusedtoensurethetreeremainsapproximatelybalanced– duringinser=onsanddele=ons

S.M.NiazArifin,PhD

AdvancedBinaryTrees

•  Red-blacktree1.  therootandallleaves(NIL)areblack2.  ifanodeisred,thenbothitschildrenareblack3.  foreachnode,allsimplepathsfromthenodeto

descendantleavescontainthesamenumberofblacknodes

S.M.NiazArifin,PhD

AdvancedBinaryTrees

•  AVLtree•  anotherspecialtypeofself-balancingBST•  heightsofthetwochildsubtreesofanynodedifferbyatmostone

•  ifatany=metheydifferbymorethanone,rebalancingisdonetorestorethisproperty

S.M.NiazArifin,PhD

AdvancedBinaryTrees

S.M.NiazArifin,PhD

Ques=ons?

S.M.NiazArifin,PhD

ThankYou!