Subject:

A Binary Search Tree (BST) is a binary tree in which each vertex has only up to 2 children that satisfies **BST property**: All vertices in the left subtree of a vertex must hold a value smaller than its own and all vertices in the right subtree of a vertex must hold a value larger than its own (we have assumption that all values are distinct integers in this visualization and small tweak is needed to cater for duplicates/non integer).

Try clicking Search(7) for a sample animation on searching a random value ∈ [1..99] in the random BST above.

An Adelson-Velskii Landis (AVL) tree is a **self-balancing** BST that maintains it's height to be O(log **N**) when having **N** vertices in the AVL tree.

Click 'Next' (on the top right)/press 'Page Down' to advance this e-Lecture slide, use the drop down list/press 'Space' to jump to a specific slide, or Click 'X' (on the bottom right)/press 'Esc' to go to Exploration mode.

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To toggle between the standard Binary Search Tree and the AVL Tree (only different behavior during Insertion and Removal of an Integer), select the respective header.

We also have URL shortcut to quickly access the AVL Tree mode, which is __https://visualgo.net/en/avl__ (you can change the 'en' to your two characters preferred language - if available).

Pro-tip: Since you are not __logged-in__, you may be a first time visitor who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: **[PageDown]** to advance to the next slide, **[PageUp]** to go back to the previous slide, **[Esc]** to toggle between this e-Lecture mode and exploration mode.

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BST (and especially balanced BST like AVL Tree) is an efficient data structure to implement *a certain kind* of **Table** (or **Map**) Abstract Data Type (ADT).

A Table ADT must support **at least** the following three operations as efficient as possible:

- Search(v) — determine if
**v**exists in the ADT or not, - Insert(v) — insert
**v**into the ADT, - Remove(v) — remove
**v**from the ADT.

Reference: See __similar slide in Hash Table e-Lecture__.

Another pro-tip: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution **or larger** (typical modern laptop resolution in 2017).

We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (**F11**) to enjoy this setup. However, you can use zoom-in (**Ctrl +**) or zoom-out (**Ctrl -**) to calibrate this.

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We are referring to Table ADT where the keys need to be ordered (as opposed to Table ADT where the keys do not need to be unordered).

This special requirement of Table ADT will be made clearer in the next few slides.

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If we use **unsorted** array/vector to implement Table ADT, it can be inefficient:

- Search(v) runs in O(
**N**), as we may end up exploring all**N**elements of the ADT if**v**actually does not exist, - Insert(v) can be implemented in O(
**1**), just put**v**at the back of the array, - Remove(v) runs in O(
**N**) too as we have to first search for**v**which is already O(**N**) and later close the resulting gap after deletion — also in O(**N**).

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If we use **sorted** array/vector to implement Table ADT, we can improve the Search(v) performance but weakens the Insert(v) performance:

- Search(v) can now be implemented in O(log
**N**), as we can now use**binary search**on the sorted array, - Insert(v) now runs in O(
**N**) as we need to implement an insertion-sort like strategy to make the array remains sorted, - Remove(v) runs in O(
**N**) because even if Search(v) runs in O(log**N**), we still need to close the gap after deletion — which is in O(**N**).

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The goal for this e-Lecture is to introduce BST and then balanced BST (AVL Tree) data structure so that we can implement the basic Table ADT operations: Search(v), Insert(v), Remove(v), **and a few other Table ADT operations** — see the next slide — in O(log **N**) time — which is much smaller than **N**.

PS: Some of the more experienced readers may notice that ∃ __another data structure__ that can implement the three basic Table ADT operations in faster time, but read on...

N | ≈ 1 000 | ≈ 1 000 000 | ≈ 1 000 000 000 |

log N | 10 | Only 20 :O | Only 30 :O:O |

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On top of the basic three, there are a few other possible Table ADT operations:

- Find the Min()/Max() element,
- Find the Successor(v) — 'next larger'/Predecessor(v) — 'previous smaller' element,
- List elements in sorted order,
- Operation X & Y - hidden for pedagogical purpose in an NUS module,
- There are others possible operations.

Discussion: What are the best possible implementation for the first three additional operations if we are limited to use [sorted|unsorted] array/vector?

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The simpler data structure that can be used to implement Table ADT is __Linked List__.

Quiz: **Can we perform all basic three Table ADT operations: Search(v)/Insert(v)/Remove(v) efficiently (read: faster than O( N)) using Linked List?**

Discussion: Why?

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Another data structure that can be used to implement Table ADT is __Hash Table__.

It has very fast Search(v), Insert(v), and Remove(v) performance (all in expected O(**1**) time).

Quiz: **So what is the point of learning this BST module if Hash Table can do the crucial Table ADT operations in unlikely-to-be-beaten expected O( 1) time?**

Discuss the answer above!

Hint: Go back to the previous 4 slides ago.

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We will now introduce BST data structure. See the visualization of an example BST above!

Root vertex does not have a parent.

There can only be one root vertex in a BST. Leaf vertex does not have any child. There can be more than one leaf vertex in a BST. Vertices that are not leaf are called the internal vertices. Sometimes root vertex is not included as part of the definition of internal vertex as the root of a BST with only one vertex can actually fit into the definition of a leaf too.

In the example above, vertex 15 is the root vertex, vertex {5, 7, 50} are the leaves, vertex {4, 6, 15 (also the root), 23, 71} are the internal vertices.

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Each vertex has **at least** 4 attributes: parent, left, right, key/value/data (there are potential other attributes).

Not all attributes will be used for all vertices, e.g. the root vertex will have its parent attribute = NULL.

## Binary Tree Visualization

Some other implementation separates key (for ordering of vertices in the BST) with the actual *satellite data* associated with the keys.

The left/right child of a vertex (except leaf) is drawn on the left/right and below of that vertex, respectively.

The parent of a vertex (except root) is drawn above that vertex. The (integer) key of each vertex is drawn inside the circle that represent that vertex. In the example above, (key) 15 has 6 as its left child and 23 as its right child. Thus the parent of 6 (and 23) is 15.

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As we do not allow duplicate integer in this visualization, the BST property is as follow: For every vertex X, all vertices on the left subtree of X are **strictly** smaller than X and all vertices on the right subtree of X are **strictly** greater than X.

In the example above, the vertices on the left subtree of the root 15: {4, 5, 6, 7} are all smaller than 15 and the vertices on the right subtree of the root 15: {23, 50, 71} are all greater than 15.

You can recursively check BST property on other vertices too.

For more complete implementation, we should consider duplicate integers too and we must consistently place integers that are equal to X to one subtree only (not arbitrary).

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We provide visualization for the following common BST/AVL Tree operations:

- Query operations (the BST structure remains unchanged):
- Search(v),
- Predecessor(v) (and similarly Successor(v)), and
- Inorder Traversal (
*we will add Preorder and Postorder Traversal soon*),

- Update operations (the BST structure may likely change):
- Insert(v),
- Remove(v), and
- Create BST (several criteria).

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There are a few other BST (Query) operations that have not been visualized in VisuAlgo:

- Rank(v): Given a key
**v**, determine what is its rank (1-based index) in the sorted order of the BST elements.That is, Rank(FindMin()) = 1 and Rank(FindMax()) =

**N**. If**v**does not exist, we can report -1. - Select(k): Given a rank
**k**, 1 ≤**k**≤**N**, determine the key**v**that has that rank**k**in the BST. Or in another word, find the**k**-th smallest element in the BST. That is, Select(1) = FindMin() and Select(**N**) = FindMax().

The details of these two operations are currently hidden for pedagogical purpose in a certain NUS module.

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Data structure that is only efficient if there is no (or rare) update, especially the insert and/or remove operation(s) is called **static** data structure.

Data structure that is efficient even if there are many update operations is called **dynamic** data structure.

BST and especially balanced BST (e.g. AVL Tree) are in this category.

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Because of the way data (distinct integers for this visualization) is organised inside a BST, we can **binary** search for an integer **v** efficiently (hence the name of **Binary Search** Tree).

First, we set the current vertex = root and then check if the current vertex is smaller/equal/larger than integer **v** that we are searching for.

## Easy Tree Diagram Generator

We then go to the right subtree/stop/go the left subtree, respectively. We keep doing this until we either find the required vertex or we don't.

On the example BST above, try clicking Search(15) (found after just 1 comparison), Search(7) (found after 3 comparisons), Search(21) (not found after 3 comparisons).

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Similarly, because of the way data is organised inside a BST, we can find the minimum/maximum element (an integer in this visualization) by starting from root and keep going to the left/right subtree, respectively.

Try clicking FindMin() and FindMax() on the example BST shown above.

The answers should be 4 and 71 (both after 4 comparisons).

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Search(v)/FindMin()/FindMax() operations run in O(**h**) where **h** is the height of the BST.

But note that this **h** can be as tall as O(**N**) in a normal BST as shown in the random 'skewed right' example above. Try Search(100) (this value should not exist as we only use random integers between [1..99] to generate this random BST and thus the Search routine should check all the way from root to the only leaf in O(**N**) time — not efficient.

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Because of the BST properties, we can find the Successor of an integer **v** (assume that we already know where integer **v** is located from earlier call of Search(**v**)) as follows:

- If
**v**has a right subtree, the minimum integer in the right subtree of**v**must be the successor of**v**.## Brunswick Corporation

Try Successor(23) (should be 50).

- If
**v**does not have a right subtree, we need to traverse the ancestor(s) of**v**until we find 'a right turn' to vertex**w**(or alternatively, until we find the first vertex**w**that is greater than vertex**v**). Once we find vertex**w**, we will see that vertex**v**is the maximum element in the left subtree of**w**. Try Successor(7) (should be 15). - If
**v**is the maximum integer in the BST,**v**does not have a successor.Try Successor(71) (should be none).

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The operations for Predecessor of an integer **v** are defined similarly (just the mirror of Successor operations).

Try the same three corner cases (but mirrored): Predecessor(6) (should be 5), Predecessor(50) (should be 23), Predecessor(4) (should be none).

At this point, stop and ponder these three Successor(v)/Predecessor(v) cases to ensure that you understand these concepts.

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Predecessor(v) and Successor(v) operations run in O(**h**) where **h** is the height of the BST.

But recall that this **h** can be as tall as O(**N**) in a normal BST as shown in the random 'skewed right' example above.

If we call Successor(FindMax()), we will go up from that last leaf back to the root in O(**N**) time — not efficient.

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We can perform an **Inorder Traversal** of this BST to obtain a list of sorted integers inside this BST (in fact, if we 'flatten' the BST into one line, we will see that the vertices are ordered from smallest/leftmost to largest/rightmost).

Inorder Traversal is a recursive method whereby we visit the left subtree first, exhausts all items in the left subtree, visit the current root, before exploring the right subtree and all items in the right subtree.

Without further ado, let's try Inorder Traversal to see it in action on the example BST above.

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Inorder Traversal runs in O(**N**), regardless of the height of the BST.

Discussion: Why?

PS: Some people call insertion of **N** unordered integers into a BST in O(**N** log **N**) and then performing the O(**N**) Inorder Traversal as '**BST sort**'.

It is rarely used though as there are several easier-to-use (comparison-based) __sorting algorithms__ than this.

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**and you are really a CS lecturer (show your University staff profile)**.

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We have not included the animation of these two other classic tree traversal methods, *but we will do so very soon*.

But basically, in Preorder Traversal, we visit the current root before going to left subtree and then right subtree.

For the example BST shown in the background, we have: {{15}, {6, 4, 5, 7}, {23, 71, 50}}.

PS: Do you notice the recursive pattern? root, members of left subtree of root, members of right subtree of root.

In Postorder Traversal, we visit the left subtree and right subtree first, before visiting the current root.

## Create/Construct Binary Search Tree (Algorithm/Program/Code)

For the example BST shown in the background, we have: {{5, 4, 7, 6}, {50, 71, 23}, {15}}.

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We can insert a new integer into BST by doing similar operation as **Search(v)**. But this time, instead of reporting that the new integer is not found, we create a new vertex in the insertion point and put the new integer there. Try Insert(60) on the example above.

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**Insert(v)** runs in O(**h**) where **h** is the height of the BST.

By now you should be aware that this **h** can be as tall as O(**N**) in a normal BST as shown in the random 'skewed right' example above.

If we call Insert(FindMax()+1), i.e.

## Binary Search Tree

we insert a new integer greater than the current max, we will go from root down to the last leaf and then insert the new integer as the right child of that last leaf in O(**N**) time — not efficient (note that we only allow up to h=9 in this visualization).

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Quiz: **Inserting integers [1,10,2,9,3,8,4,7,5,6] one by one in that order into an initially empty BST will result in a BST of height:**

Pro-tip: You can use the 'Exploration mode' to verify the answer.

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We can remove an integer in BST by performing similar operation as **Search(v)**.

If **v** is not found in the BST, we simply do nothing.

If **v** is found in the BST, we do not report that the existing integer **v** is found, but instead, we perform one of the three possible removal cases that will be elaborated in three separate slides (we suggest that you try each of them one by one).

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The first case is the easiest: Vertex **v** is currently one of the leaf vertex of the BST.

Deletion of a leaf vertex is very easy: We just remove that leaf vertex — try Remove(5) on the example BST above (second click onwards after the first removal will do nothing — please refresh this page or go to another slide and return to this slide instead).

This part is clearly O(**1**) — on top of the earlier O(**h**) search-like effort.

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The second case is also not that hard: Vertex **v** is an (internal/root) vertex of the BST and it has **exactly one child**.

Removing **v** without doing anything else will disconnect the BST.

Deletion of a vertex with one child is not that hard: We connect that vertex's only child with that vertex's parent — try Remove(23) on the example BST above (second click onwards after the first removal will do nothing — please refresh this page or go to another slide and return to this slide instead).

This part is also clearly O(**1**) — on top of the earlier O(**h**) search-like effort.

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The third case is the most complex among the three: Vertex **v** is an (internal/root) vertex of the BST and it has **exactly two children**.

Removing **v** without doing anything else will disconnect the BST.

Deletion of a vertex with two children is as follow: We replace that vertex with its successor, and then delete its duplicated successor in its right subtree — try Remove(6) on the example BST above (second click onwards after the first removal will do nothing — please refresh this page or go to another slide and return to this slide instead).

This part requires O(**h**) due to the need to find the successor vertex — on top of the earlier O(**h**) search-like effort.

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This case 3 warrants further discussions:

- Why replacing a vertex
**B**that has two children with its successor**C**is always a valid strategy? - Can we replace vertex
**B**that has two children with its predecessor**A**instead?Why or why not?

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**and you are really a CS lecturer (show your University staff profile)**.

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**Remove(v)** runs in O(**h**) where **h** is the height of the BST. Removal case 3 (deletion of a vertex with two children is the 'heaviest' but it is not more than O(**h**)).

As you should have fully understand by now, **h** can be as tall as O(**N**) in a normal BST as shown in the random 'skewed right' example above.

If we call Remove(FindMax()), i.e.

## Tree Diagram Maker

we remove the current max integer, we will go from root down to the last leaf in O(**N**) time before removing it — not efficient.

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To make life easier in 'Exploration Mode', you can create a new BST using these options:

**Empty**BST (you can then insert a few integers one by one),- The two
**e-Lecture Examples**that you may have seen several times so far, **Random**BST (which unlikely to be extremely tall),**Skewed Left/Right**BST (tall BST with**N**vertices and**N-1**linked-list like edges, to showcase the worst case behavior of BST operations; disabled in AVL Tree mode).

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We are midway through the explanation of this BST module.

So far we notice that many basic Table ADT operations run in O(**h**) and **h** can be as tall as **N-1** edges like the 'skewed left' example shown — inefficient :(...

So, is there a way to make our BSTs 'not that tall'?

PS: If you want to study how these basic BST operations are implemented in a real program, you can download this __BSTDemo.cpp__.

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At this point, we encourage you to press [Esc] or click the X button on the bottom right of this e-Lecture slide to enter the 'Exploration Mode' and try various BST operations yourself to strengthen your understanding about this versatile data structure.

When you are ready to continue with the explanation of **balanced** BST (we use **AVL Tree** as our example), press [Esc] again or switch the mode back to 'e-Lecture Mode' from the top-right corner drop down menu.

Then, use the slide selector drop down list to resume from __this slide 12-1__.

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We have seen from earlier slides that most of our BST operations except Inorder traversal runs in O(**h**) where **h** is the height of the BST that can be as tall as **N-1**.

We will continue our discussion with the concept of **balanced BST** so that **h** = O(log **N**).

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There are several known implementations of balanced BST, too many to be visualized and explained one by one in VisuAlgo.

We focus on **AVL Tree** (Adelson-Velskii & Landis, 1962) that is named after its inventor: Adelson-Velskii and Landis.

Other balanced BST implementations (more or less as good or slightly better in terms of constant-factor performance) are: Red-Black Tree, B-trees/2-3-4 Tree (Bayer & McCreight, 1972), Splay Tree (Sleator and Tarjan, 1985), Skip Lists (Pugh, 1989), Treaps (Seidel and Aragon, 1996), etc.

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To facilitate AVL Tree implementation, we need to **augment** — add more information/attribute to — each BST vertex.

For each vertex **v**, we define **height(v)**: The number of edges on the path from vertex **v** down to its deepest leaf.

This attribute is saved in each vertex so we can access a vertex's height in O(**1**) without having to recompute it every time.

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Formally:

v.height = -1 (if v is an empty tree)v.height = max(v.left.height, v.right.height) + 1 (otherwise)The height of the BST is thus: .

On the example BST above, height(11) = height(32) = height(50) = height(72) = height(99) = 0 (all are leaves).

height(29) = 1 as there is 1 edge connecting it to its only leaf 32.

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If we have **N** elements/items/keys in our BST, the lower bound height **h** > log_{2}**N** if we can somehow insert the **N** elements in perfect order so that the BST is perfectly balanced.

See the example shown above for **N = 15** (a perfect BST which is rarely achievable in real life — try inserting any other integer and it will not be perfect anymore).

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+ 2^{h}

N ≤ 2^{0} + 2^{1} + 2^{2} + … + 2^{h}

N < 2^{h+1} (sum of geometric progression)

log_{2} N < log_{2} 2^{h+1}

log_{2} N < (h+1) * log_{2} 2 (log_{2} 2 is 1)

h > (log_{2} N)-1 (algebraic manipulation)

h > log_{2} N

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If we have **N** elements/items/keys in our BST, the upper bound height **h** < **N** if we insert the elements in ascending order (to get skewed right BST as shown above).

The height of such BST is , so we have .

Discussion: Do you know how to get skewed left BST instead?

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**and you are really a CS lecturer (show your University staff profile)**.

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We have seen that most BST operations are in O(**h**) and combining the lower and upper bounds of **h**, we have log_{2}**N** < **h** < **N**.

There is a dramatic difference between log_{2}**N** and **N** and we have seen from the discussion of the lower bound that getting perfect BST (at all times) is near impossible...

So can we have BST that has height closer to log_{2}**N**, i.e.

**c** * log_{2}**N**, for a small constant factor **c**?

If we can, then BST operations that run in O(**h**) actually run in O(log **N**)...

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Introducing AVL Tree, invented by two Russian (Soviet) inventors: Georgy Adelson-Velskii and Evgenii Landis, back in 1962.

In AVL Tree, we will later see that its height **h** < 2 * log **N** (tighter analysis exist, but we will use easier analysis in VisuAlgo where **c = 2**).

Therefore, most AVL Tree operations run in O(log **N**) time — efficient.

Insert(v) and Remove(v) update operations may change the height **h** of the AVL Tree, but we will see **rotation** operation(s) to maintain the AVL Tree height to be low.

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To have efficient performance, we shall not maintain **height(v)** attribute via the O(**N**) recursive method every time there is an update (Insert(v)/Remove(v)) operation.

Instead, we compute O(**1**): at the back of our Insert(v)/Remove(v) operation as only the height of vertices along the insertion/removal path may be affected.

Thus, only O(**h**) vertices may change its **height(v)** attribute and in AVL Tree, **h** < 2 * log **N**.

Try Insert(37) on the example AVL Tree (ignore the resulting rotation for now, we will come back to it in the next few slides).

Notice that only a few vertices along the insertion path: {41,20,29,32} increases their height by +1 and all other vertices will have their heights unchanged.

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Let's define the following important AVL Tree invariant (property that will never change): A vertex **v** is said to be **height-balanced** if .

A BST is called height-balanced according to the invariant above if every vertex in the BST is height-balanced.

## Transform Your Enterprise Platforms and Cloud Migrations

Such BST is called AVL Tree, like the example shown above.

Take a moment to pause here and try inserting a few new random vertices or deleting a few random existing vertices. Will the resulting BST still considered height-balanced?

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Adelson-Velskii and Landis claim that an AVL Tree (a height-balanced BST that satisfies AVL Tree invariant) with **N** vertices has height **h** < 2 * log_{2}**N**.

The proof relies on the concept of minimum-size AVL Tree of a certain height **h**.

Let **N _{h}** be the minimum number of vertices in a height-balanced AVL Tree of height

**h**.

The first few values of **N _{h}** are

**N**(a single root vertex),

_{0}= 1**N**(a root vertex with either one left child or one right child only),

_{1}= 2**N**,

_{2}= 4**N**,

_{3}= 7**N**,

_{4}= 12**N**(see the background picture), and so on (see the next two slides).

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We know that for any other AVL Tree of **N** vertices (not necessarily the minimum-size one), we have **N ≥ N _{h}**.

In the background picture, we have **N _{5} = 20** vertices but we know that we can squeeze 43 more vertices (up to

**N = 63**) before we have a perfect binary tree of height

**h = 5**.

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_{h}= 1 + N

_{h-1}+ N

_{h-2}(formula for minimum-size AVL tree of height h)

N

_{h}> 1 + 2*N

_{h-2}(as N

_{h-1}> N

_{h-2})

N

_{h}> 2*N

_{h-2}(obviously)

N

_{h}> 4*N

_{h-4}(recursive)

N

_{h}> 8*N

_{h-6}(another recursive step)

...

(we can only do this h/2 times, assuming initial h is even)

N_{h} > 2^{h/2}*N_{0} (we reach base case)

N_{h} > 2^{h/2} (as N_{0} = 1)

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_{h}> 2

^{h/2}(combining the previous two slides)

N > 2

^{h/2}

log

_{2}(N) > log

_{2}(2

^{h/2}) (log

_{2}on both sides)

log

_{2}(N) > h/2 (formula simplification)

2 * log

_{2}(N) > h or h < 2 * log

_{2}(N)

h = O(log(N)) (the final conclusion)

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Look at the example BST again.

See that all vertices are height-balanced, an AVL Tree.

To quickly detect if a vertex **v** is height balanced or not, we modify the AVL Tree invariant (that has absolute function inside) into: .

Now try Insert(37) on the example AVL Tree again.

A few vertices along the insertion path: {41,20,29,32} increases their height by +1. Vertices {29,20} will no longer be height-balanced after this insertion (and will be rotated later — discussed in the next few slides), i.e.

bf(29) = -2 and bf(20) = -2 too. We need to restore the balance.

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See the picture above. Calling **rotateRight(Q)** on the left picture will produce the right picture. Calling **rotateLeft(P)** on the right picture will produce the left picture again.

**rotateRight(T)**/**rotateLeft(T)** can only be called if **T** has a left/right child, respectively.

Tree Rotation **preserves** BST property.

## Cryptocurrency New In 2018

Before rotation, P ≤ B ≤ Q. After rotation, notice that subtree rooted at B (if it exists) changes parent, but P ≤ B ≤ Q does not change.

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BSTVertex w = T.right // rotateRight is the mirror copy of this

w.parent = T.parent // this method is hard to get right for newbie

T.parent = w

T.right = w.left

if (w.left != null) w.left.parent = T

w.left = T

// update the height of T and then w here

return w

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Basically, there are only these four imbalance cases.

We use Tree Rotation(s) to deal with each of them.

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- Just insert
**v**as in normal BST, - Walk up the AVL Tree from the insertion point back to the root and at every step, we update the height and balance factor of the affected vertices:
- Stop at the
**first**vertex that is out-of-balance (+2 or -2), if any, - Use
**one**of the four tree rotation cases to rebalance it again, e.g.try Insert(37) on the example above and notice by calling

**rotateLeft(29)**once, we fix the imbalance issue.

- Stop at the

Discussion: Is there other tree rotation cases for Insert(v) operation of AVL Tree?

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**and you are really a CS lecturer (show your University staff profile)**.

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- Just remove
**v**as in normal BST (one of the three removal cases), - Walk up the AVL Tree from the deletion point back to the root and at every step, we update the height and balance factor of the affected vertices:
- Now for
**every**vertex that is out-of-balance (+2 or -2), we use**one**of the four tree rotation cases to rebalance**them**(can be more than one) again.

- Now for

The main difference compared to Insert(v) in AVL tree is that we may trigger one of the four possible rebalancing cases **several times**, but not more than **h** = O(log **N**) times :O, try Remove(7) on the example above to see two chain reactions **rotateRight(6)** and then **rotateRight(16)+rotateLeft(8)** combo.

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We have now see how AVL Tree defines the height-balance invariant, maintain it for all vertices during Insert(v) and Remove(v) update operations, and a proof that AVL Tree has **h** < 2 * log **N**.

Therefore, **all** BST operations (both update and query operations except Inorder Traversal) that we have learned so far, if they have time complexity of O(**h**), they have time complexity of O(log **N**) if we use AVL Tree version of BST.

This marks the end of this e-Lecture, but please switch to 'Exploration Mode' and try making various calls to Insert(v) and Remove(v) in AVL Tree mode to strengthen your understanding of this data structure.

PS: If you want to study how these seemingly complex AVL Tree (rotation) operations are implemented in a real program, you can download this __AVLDemo.cpp__ (must be used together with this __BSTDemo.cpp__).

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We will end this module with a few more interesting things about BST and balanced BST (especially AVL Tree).

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**and you are really a CS lecturer (show your University staff profile)**.

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## Binary tree creator online

**and you are really a CS lecturer (show your University staff profile)**.

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For a few more interesting questions about this data structure, please practice on __BST/AVL__ training module (no login is required).

However, for registered users, you should login and then go to the __Main Training Page__ to officially clear this module and such achievement will be recorded in your user account.

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We also have a few programming problems that somewhat requires the usage of this **balanced** BST (like AVL Tree) data structure: __Kattis - compoundwords__ and __Kattis - baconeggsandspam__.

Try them to consolidate and improve your understanding about this data structure.

You are allowed to use C++ STL map/set or Java TreeMap/TreeSet if that simplifies your implementation.

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**and you are really a CS lecturer (show your University staff profile)**.

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As the action is being carried out, each step will be described in the status panel.

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**and you are really a CS lecturer (show your University staff profile)**.

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Control the animation with the player controls!

Keyboard shortcuts are:

**Spacebar:** play/pause/replay

**Left/right arrows:**step backward/step forward

**-/+:**decrease/increase speed

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Return to 'Exploration Mode' to start exploring!

Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com.

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Create

Search(v)

Insert(v)

Remove(v)

Pred-/Succ-essor(v)

Inorder Traversal

AboutTeamTerms of use

#### About

✕VisuAlgo was conceptualised in 2011 by Dr Steven Halim as a tool to help his students better understand data structures and algorithms, by allowing them to learn the basics on their own and at their own pace.

VisuAlgo contains many advanced algorithms that are discussed in Dr Steven Halim's book ('Competitive Programming', co-authored with his brother Dr Felix Halim) and beyond.

Today, some of these advanced algorithms visualization/animation can only be found in VisuAlgo.

Though specifically designed for National University of Singapore (NUS) students taking various data structure and algorithm classes (e.g.

CS1010, CS1020, CS2010, CS2020, CS3230, and CS3230), as advocators of online learning, we hope that curious minds around the world will find these visualisations useful too.

VisuAlgo is not designed to work well on small touch screens (e.g. smartphones) from the outset due to the need to cater for many complex algorithm visualizations that require lots of pixels and click-and-drag gestures for interaction. The minimum screen resolution for a respectable user experience is 1024x768 and only the landing page is relatively mobile-friendly.

VisuAlgo is an ongoing project and more complex visualisations are still being developed.

The most exciting development is the automated question generator and verifier (the online quiz system) that allows students to test their knowledge of basic data structures and algorithms.

The questions are randomly generated via some rules and students' answers are instantly and automatically graded upon submission to our grading server. This online quiz system, when it is adopted by more CS instructors worldwide, should technically eliminate manual basic data structure and algorithm questions from typical Computer Science examinations in many Universities. By setting a small (but non-zero) weightage on passing the online quiz, a CS instructor can (significantly) increase his/her students mastery on these basic questions as the students have virtually infinite number of training questions that can be verified instantly before they take the online quiz.

The training mode currently contains questions for 12 visualization modules. We will soon add the remaining 8 visualization modules so that every visualization module in VisuAlgo have online quiz component.

Another active branch of development is the internationalization sub-project of VisuAlgo. We want to prepare a database of CS terminologies for all English text that ever appear in VisuAlgo system. This is a big task and requires crowdsourcing. Once the system is ready, we will invite VisuAlgo visitors to contribute, especially if you are not a native English speaker.

Currently, we have also written public notes about VisuAlgo in various languages: __zh__, __id__, __kr__, __vn__, __th__.

#### Team

✕**Project Leader & Advisor (Jul 2011-present)**

Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)

Dr Felix Halim, Software Engineer, Google (Mountain View)

**Undergraduate Student Researchers 1 (Jul 2011-Apr 2012)**

Koh Zi Chun, Victor Loh Bo Huai

**Final Year Project/UROP students 1 (Jul 2012-Dec 2013)**

Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy

**Final Year Project/UROP students 2 (Jun 2013-Apr 2014)**

Rose Marie Tan Zhao Yun, Ivan Reinaldo

**Undergraduate Student Researchers 2 (May 2014-Jul 2014)**

Jonathan Irvin Gunawan, Nathan Azaria, Ian Leow Tze Wei, Nguyen Viet Dung, Nguyen Khac Tung, Steven Kester Yuwono, Cao Shengze, Mohan Jishnu

**Final Year Project/UROP students 3 (Jun 2014-Apr 2015)**

Erin Teo Yi Ling, Wang Zi

**Final Year Project/UROP students 4 (Jun 2016-Dec 2017)**

Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

List of translators who have contributed ≥100 translations can be found at statistics page.

**Acknowledgements**

This project is made possible by the generous Teaching Enhancement Grant from NUS Centre for Development of Teaching and Learning (CDTL).

#### Terms of use

✕VisuAlgo is free of charge for Computer Science community on earth.

If you like VisuAlgo, the only payment that we ask of you is for you to **tell the existence of VisuAlgo to other Computer Science students/instructors** that you know =) via Facebook, Twitter, course webpage, blog review, email, etc.

If you are a data structure and algorithm **student/instructor**, you are allowed to use this website directly for your classes.

If you take screen shots (videos) from this website, you can use the screen shots (videos) elsewhere as long as you cite the URL of this website (http://visualgo.net) and/or list of publications below as reference. However, you are **NOT** allowed to download VisuAlgo (client-side) files and **host it** on your own website as it is **plagiarism**.

As of now, we do **NOT** allow other people to fork this project and create variants of VisuAlgo. Using the offline copy of (client-side) VisuAlgo for your personal usage is fine.

Note that VisuAlgo's online quiz component is by nature has heavy server-side component and there is no easy way to save the server-side scripts and databases locally. Currently, the general public can only use the 'training mode' to access these online quiz system.

Currently the 'test mode' is a more controlled environment for using these randomly generated questions and automatic verification for a **real** examination in NUS. Other interested CS instructor should contact Steven if you want to try such 'test mode'.

**List of Publications**

This work has been presented briefly at the CLI Workshop at the ACM ICPC World Finals 2012 (Poland, Warsaw) and at the IOI Conference at IOI 2012 (Sirmione-Montichiari, Italy).

You can click this link to read our 2012 paper about this system (it was not yet called VisuAlgo back in 2012).

This work is done mostly by my past students. The most recent final reports are here: Erin, Wang Zi, Rose, Ivan.

**Bug Reports or Request for New Features**

VisuAlgo is not a finished project.

Dr Steven Halim is still actively improving VisuAlgo. If you are using VisuAlgo and spot a bug in any of our visualization page/online quiz tool or if you want to request for new features, please contact Dr Steven Halim.

His contact is the concatenation of his name and add gmail dot com.