Binary Search Trees (BST) Explained in Animated Demo

Summary

In this video, Joe James explains binary search trees, a data structure with nodes connected by edges. Each node can have up to two children, forming a hierarchy with a root. Binary search trees maintain order, with each node greater than its left subtree and less than its right. Operations like insertion, deletion, and finding a node are efficient, taking O(h) time, where h is the height of the tree. The video also covers how to handle deletions with different child counts, emphasizing the speed benefits of balanced trees.

Takeaways

• 🌳 A binary search tree (BST) is a hierarchical structure where each node has at most two child nodes, arranged in a way that the left child node's value is less than the parent node's value, and the right child node's value is greater.
• 🔍 In a BST, searching for a node starts at the root and compares the target value with the current node's value, moving to the left or right child accordingly until the node is found or the target is determined to not exist in the tree.
• 🍃 Leaf nodes in a BST are nodes that do not have any children.
• 📚 The left and right children of a node are referred to as siblings.
• 📈 The property of a BST ensures that for any given node, all nodes in its left subtree are less than the node and all nodes in its right subtree are greater.
• 💼 Insertion in a BST involves starting at the root and moving to the left or right child depending on whether the value to insert is less than or greater than the current node's value until an appropriate position to insert the new node as a leaf is found.
• 🔑 The find operation in a BST is efficient because it eliminates half of the remaining tree with each comparison, making it much faster than a linear search.
• ❌ Deletion in a BST can be complex and depends on whether the node to be deleted is a leaf, has one child, or two children, each requiring a different approach to maintain the BST properties.
• 🔄 If a node to be deleted has one child, the child replaces the node in the tree, effectively removing the node without disrupting the BST properties.
• 🔄 If a node to be deleted has two children, the typical approach is to find the in-order successor (the smallest node in the right subtree) or the in-order predecessor (the largest node in the left subtree), replace the node with this successor or predecessor, and then delete the successor or predecessor.
• 🏃 The operations of insertion, deletion, and find in a BST are generally O(h), where h is the height of the tree, making them very efficient in balanced trees.

Q & A

What is a binary search tree?

A binary search tree is a binary tree data structure where each node has at most two children, which are referred to as the left child and the right child. For each node, all elements in its left subtree are less than the node and all elements in its right subtree are greater.

What is the significance of the root node in a binary search tree?

The root node is the topmost node in the tree and serves as the starting point for operations such as insertion, deletion, and searching.

What are the characteristics of leaf nodes in a binary search tree?

Leaf nodes are nodes that do not have any children. They are the terminal nodes in the tree structure.

How are siblings defined in a binary search tree?

Siblings in a binary search tree are nodes that share the same parent node.

What is the rule for ordering nodes in a binary search tree?

In a binary search tree, each node is greater than every node in its left subtree and less than every node in its right subtree.

How does the insert operation work in a binary search tree?

The insert operation starts at the root and compares the value to be inserted with the current node's value. Depending on whether the value is less than or greater than the current node's value, it moves to the left or right child, respectively, until it finds the correct position to insert the new node as a leaf.

Can you explain the find operation in a binary search tree?

The find operation starts at the root and compares the target value with the current node's value. It traverses the tree, moving to the left child if the target is less and to the right child if the target is greater, until it either finds the node or reaches a null child, indicating the node is not present.

What are the three cases for the delete operation in a binary search tree?

The three cases for the delete operation are: 1) Deleting a leaf node, which can be simply removed. 2) Deleting a node with one child, where the child replaces the node. 3) Deleting a node with two children, where the node is replaced with its in-order successor or predecessor, usually the smallest value in its right subtree or the largest in its left subtree.

Why is the next highest node used when deleting a node with two children?

The next highest node (in-order successor) is used to maintain the binary search tree property after deletion. It replaces the deleted node and is either the smallest node in the right subtree (or the largest in the left subtree), ensuring that all nodes in the left subtree are still less and all in the right subtree are still greater.

What is the time complexity for insert, find, and delete operations in a binary search tree?

The time complexity for insert, find, and delete operations in a binary search tree is O(h), where h is the height of the tree. This means that in a balanced tree, these operations can be performed very efficiently, often in logarithmic time.

Why are binary search trees considered efficient for certain operations?

Binary search trees are considered efficient because they allow for fast lookup, addition, and removal of items. In a balanced tree, these operations can be performed in O(log n) time, where n is the number of nodes in the tree.

Outlines

00:00

🌳 Introduction to Binary Search Trees

Joe James introduces binary search trees (BSTs), explaining that a tree is a collection of nodes connected by edges, organized into levels with a root node at the top. Each node can have up to two child nodes, referred to as left and right children. Leaf nodes are those without children. The video describes the concept of siblings, ancestors, and descendants within the tree structure. It then defines the properties of BSTs, where each node is greater than all nodes in its left subtree and less than all nodes in its right subtree. The operations of insertion, finding, and deletion are outlined, with an example of inserting the value '12' into the tree. The process starts at the root and compares the value to be inserted with the current node, moving left or right accordingly until the correct position is found to insert the new node as a leaf.

05:00

🔍 Operations and Efficiency of BSTs

The second paragraph delves into the operations of BSTs, specifically focusing on deletion. Three cases are presented: deleting a leaf node, a node with one child, and a node with two children. For leaf nodes, deletion is straightforward as it does not affect the rest of the tree. For nodes with one child, the child replaces the node in the parent's structure. For nodes with two children, the 'next higher node' is found and replaces the node to be deleted, after which the higher node is deleted. This process is done recursively. The paragraph concludes by emphasizing the speed advantage of BSTs, where operations like insert, delete, and find can be performed in O(h) time, where h is the height of the tree. This results in very fast operations, especially in a balanced BST with millions of nodes, requiring only a few comparisons. The video wraps up with a summary of the key points and a thank you from Joe James.

Keywords

💡Binary Search Tree (BST)

A Binary Search Tree is a data structure in which each node has at most two child nodes: a left child and a right child. The key property of a BST is that each node is greater than all nodes in its left subtree and less than all nodes in its right subtree. This structure is crucial for efficient searching, insertion, and deletion, which are performed based on node comparisons, as described in the video.

💡Node

A node is a fundamental part of a tree structure in computer science. It represents an individual element in the tree and may contain a value or data. Each node can connect to child nodes through edges, forming parent-child relationships. For example, in the video, the node labeled '5' has both a left child ('3') and a right child ('8').

💡Root Node

The root node is the top-most node in a tree structure, and all other nodes are descendants of it. In the video, node '15' is used as an example of a root node. It is where operations such as insertion and searching typically start.

💡Leaf Node

A leaf node is a node that does not have any child nodes, meaning it is at the bottom of a tree. For example, in the video, nodes such as '2', '6', '12', and '19' are leaf nodes because they do not branch further. Leaf nodes are significant in deletion operations because they can be removed without affecting the structure of the tree.

💡Subtree

A subtree is a portion of a tree that includes a node and all of its descendants. In the video, the concept of a subtree is illustrated by describing node '5' and its left and right subtrees, which include all the nodes beneath '5'. Subtrees are critical in understanding operations like insertion and deletion.

💡Ancestor

An ancestor of a node in a tree includes any node that exists along the path from that node to the root. For example, in the video, node '4' has ancestors '3', '5', and '15'. Understanding ancestors is important when traversing a tree to perform operations such as insertion or deletion.

💡Descendant

A descendant is any node that exists in the subtree of another node. In the video, all the nodes below '5' are referred to as its descendants. This concept helps in defining relationships between nodes during tree operations.

💡Insert Operation

Insertion in a binary search tree involves adding a new node in such a way that the BST properties are preserved. As described in the video, an insertion starts at the root node and proceeds by comparing the new value with the current nodes until the appropriate leaf position is found for the new node. For example, inserting '12' into the tree required several comparisons before placing it as a child of node '13'.

💡Find Operation

The find operation in a binary search tree is a process used to locate a specific node based on its value. The video explains that to find node '19', the search begins at the root and compares the target value to nodes while traversing the tree. This operation takes advantage of the BST's ordered structure for efficiency.

💡Delete Operation

The delete operation in a binary search tree removes a node, with three cases to handle: when the node is a leaf, has one child, or has two children. The video details how, in each case, the tree is restructured to maintain its properties. For example, when deleting node '24', the next highest node ('25') is used to replace it.

Highlights

A tree is a collection of nodes with edges that connect them.

Nodes are organized into levels with the top level called the root node.

Each node can have child nodes, including left and right children.

Siblings are nodes that share the same parent.

Leaf nodes are nodes without any children.

In a binary tree, each node can have up to two child nodes.

Node five's right subtree includes everything under node five.

Node four has ancestors, which are every node between itself and the root node.

Node five has descendants, which are all nodes below it.

In a binary search tree, each node is greater than every node in its left subtree.

Each node in a binary search tree is less than every node in its right subtree.

Binary search tree operations include insert, find, and delete.

For insertion, start at the root and compare the value to be inserted.

The find operation starts at the root and compares the target value.

The delete operation has three cases: leaf node, one child, or two children.

If a node to be deleted is a leaf, it can be simply removed.

If a node has one child, the child takes the node's place in the tree.

If a node has two children, find the next highest node and replace the node to be deleted.

Binary search trees offer fast insertion, deletion, and find operations with Big O of H speed.

In a balanced binary search tree, operations can be incredibly fast with minimal comparisons.

Transcripts

hi I'm Joe James and this video is going

to cover binary search trees so a tree

is a collection of nodes with edges that

connect them the nodes are organized

into levels and the top level is called

the root node each node can have child

nodes in this case the three is a parent

node and it has a left child and a right

child and the left and right child are

called siblings because they share the

same parents the 1 4 6 and nine nodes

here are called Leaf nodes because they

don't have any child nodes in a binary

tree each node can have up to two child

nodes a left child and a right child

node five's right sub tree includes

everything in this triangle which is a

tree in

itself under node five node 5's left sub

tree includes noes 3 1 and four

node four has ancestors which are simply

every node between itself and the root

node and node five has descendants but

basically every node below it all of

this children node and their children's

nodes and so on in a binary search tree

each node is greater than every node in

its left sub tree so for example 15 is

greater than every node in the left sub

tree node 8 is greater than every node

in in its left sub tree node 5 is

greater than the two in its left sub

tree node 24 is greater than 19 in its

left sub tree and so on and each node is

less than every node in its right sub

tree so here we can see that node 15 is

less than all the members of its right

sub tree 24 is less than the members of

its right sub tree 8 is less than the

members of its right sub tree and so on

these two characteristics make a binary

search tree

binary search tree operations we have

insert find and delete so for a binary

search tree insertion we start at the

root which is node 15 let's say we want

to insert a 12 we're always going to

insert a node as a leaf so our first

comparison will compare 12 to 15 12 is

less than 15 we take the left child We

compare 12 to 8 12 is greater than 8 so

we take the right child We compare 12 12

to 11 12 is greater than 11 so we take

the right child we'll compare 12 to 13

and 12 is less than 13 13 has no

children so we add 12 as a new left

child for note 13 so we can see that

node 12 is inserted as a leaf node for

the binary search tree find operation we

also start at the root let's say we want

to find node 19 our first comparison is

19 less than 15

no it is not so we take the right sube

is 19 less than 24 yes it is and there

we found 19

already so find operation is pretty

simple and the binary search tree delete

operation there are three possible cases

either the node you want to delete is a

leaf node it has one child or it has two

children and we'll look at each case

separately so first let's say we want to

delete a node that is a leaf

node now in this tree we can see that

nodes 2 6 12 19 and 25 are all Leaf

nodes so if we want to delete one we can

simply delete it without affecting the

rest of the tree now if we want to

delete a node that has one child and in

our tree example here we can see that

nodes 11 13 28 all have one child if we

want to delete for example node 11 we

can see that 11 is node 8's right child

so we make 11's only child 13 eight

right child in other words 13 takes 11's

place and then we can delete node

11 now note here that node 12 stays

attached to this 13's left child and

that does not change it does move up a

level in the tree but it stays attached

as 13's left child so 13's subtree is

not

affected now if we want to do a delete

operation with two children let's say we

want to delete node 24 which we can see

has a left child and a right child we

find the next higher node so we do that

we look at node 24 we follow the right

subtree node 28 and then we look at left

child and left child continue down the

leftmost path until we get to the bottom

in this case there's only one Edge to

node 25 so that is the next highest node

from 24 so in this case we can change

the 24 to 25 and delete note 25 and last

case let's say we want to delete a node

that has two children in this case node

four so four has left child and a right

child so first we're going to find the

next higher node which is six we changed

a four to a six and then we can delete

node 6 and since the delete operation is

done recursively the seven basically

takes six's place and the six is

gone so the advantage of binary search

trees

speed we can insert delete and find in

Big O of H speed which is the Big O of

the height of the tree that means that

in a balanced binary search tree with 10

million nodes find and certain delete

operations take only on the order of 30

comparisons which is incredibly fast

that wraps up our introduction to binary

search trees I'm Joe James thanks for

watching