Binary Search Trees (BST) Explained in Animated Demo
Summary
TLDRIn 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
🌳 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.
🔍 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.
Mindmap
Keywords
💡Binary Search Tree (BST)
💡Node
💡Root Node
💡Leaf Node
💡Subtree
💡Ancestor
💡Descendant
💡Insert Operation
💡Find Operation
💡Delete Operation
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
5.0 / 5 (0 votes)