Quick but thorough run-through of binary search tree terminology: root, leaves, internal nodes, subtrees, depth, height, ancestors, descendants, siblings, left/right child – everything clearly labeled on a working example.

Great for beginners, interview prep, or reviewing foundational BST concepts before coding insert/search/delete.

00:00 Introduction to BST Terminology
00:28 Root Node
01:10 Ancestors and Descendants
01:58 Children, Grandchildren, and Siblings
04:07 Internal Nodes vs External Nodes (Leaves)
05:34 Understanding Subtrees
06:09 Left Subtree and Right Subtree Examples
08:34 Depth of a Node
11:02 Height of the Tree
12:48 Height of Subtrees
17:32 Node Structure and Pointers Overview
18:12 Closing Remarks and Call to Action

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Hello there! Let’s talk about binary search tree terminology. If you saw my

previous video we talked about how to define a binary search tree meaning a

whole bunch of rules so that if the thing you’re looking at follows all the

rules then you know you’re actually looking at a binary search tree if not

then not. So see my previous video if you want to know for sure whether you’re

looking at a binary search tree. For now we’re just going to talk about some

Okay, so the first thing we should probably obviously talk about is the root node over here.

So I mean, well, in my previous video, we talked about nodes, right?

So this is kind of a graph with a whole bunch of rules on top of it.

That means we have nodes and edges.

So if you look at the very top node here, sometimes also referred to as a vertex,

I think in binary search tree terminology, we usually say nodes.

We usually say nodes. I can’t actually remember. But so look at the 42 there

That node is the root node of all other nodes. It’s the highest common ancestor in the entire tree. So this is the root node

First bit of terminology. Also, I tried to sneak past you ancestor

So these trees are supposed to be written in a way that kind of looks like they have a family hierarchy with one parent per

like not two children per parent, but just like one parent and then either zero or one or two

children. So we’ll say that ancestors are higher on the tree. So that means 42 is actually an

ancestor of 33. And it’s also an ancestor of 12. It’s also an ancestor of 19. Just anything that’s

higher is an ancestor of anything that’s lower. We could also say that 33 is an ancestor of 39

and 19 and so forth. You can probably also imagine that we have children and grandchildren. Yeah,

we do go that far in binary search trees. So the 42 node, it has two children. It has a left child

and a right child. The 33 is the left child. I’ll put LC for left child. And the 67 is its right

child. I’ll put an RC there. The 67 in turn has two children. The 56 has no children of its own.

the 33 has two children the 12 only has one child it was you know playing it safe i guess

you never know if these children are going to come out and just like run amok

and and engage in constant shenanigans so the 12 has a right child but no left child that’s okay

um in terms of going higher on the tree anything that is higher is an ancestor sorry i should have

said lower anything that’s higher is an ancestor anything that’s lower is a descendant so if we’re

If we’re looking at the 33 node, the 33 is a descendant of 42 because it’s the left child of 42.

It’s also an ancestor of anything that comes below it.

So it’s an ancestor of 12 and 39 and 19, right?

So if we’re looking at 33, we’ve got a left child over here and we’ve got a right child over here.

And then we have a grandchild, which is the 19 node.

We don’t really have left and right grandchildren.

You could say a grandchild in the left subtree or the right subtree.

subtree talk about sub trees in a second the 33 has a parent node which is just the 42 node

which is also the root node of course it’s got a sibling the 67 is the sibling you can tell

something’s a sibling because it’s got the same parent as you it’s the people that you’re usually

fighting with right anyway so if we’re looking at any node in particular it might have a whole bunch

of ancestors above the tree it might have a whole bunch of descendants below the tree

It has siblings or it usually has zero or one sibling because in a binary search tree,

we can only have up to two children per node.

It’s got sometimes, you know, grandparents and great grandparents and children and great

grandchildren.

So just think about the hierarchy like a family tree would have.

Okay, moving on to some more terminology.

Next thing is we have internal nodes and also external nodes.

So what do I mean by internal? Internal means a node has more than zero children. It has one or two children.

So I’m going to put internal on the 33 because the 33 node has children.

The 12 also has children, so it’s internal. The 39 does not have children, so it’s not internal.

67 has children, so it’s internal. And the root node, 42, also has children, so it’s considered internal.

That 42 has a lot of different names.

It’s the root node, it’s the greatest common ancestor,

it’s an internal node and so forth.

Notice how the other nodes that I have not highlighted,

they have zero children.

So when a node has zero children,

it’s known as an external node.

It’s also known as a leaf

because we’re talking about trees

and I guess it’s kind of like a nice synonym.

So the 19, the node with no children of its own is a leaf.

So is the 39.

So is the 56.

So is the 76.

I just want to point out also, if you were with me on my last video,

then the numbers need to be ordered from left to right.

But don’t worry, we’re going to do another video where we build a complete tree from scratch.

There’s some more terminology we should talk about.

So I’m going to get rid of all these externals and internals real fast.

Or the labels.

We should talk about the left subtree versus the right subtree.

left subtree versus the right subtree i mean what is a subtree anyway the subtree is basically

a subtree is basically just pick any node you want in the entire tree let’s pick

the 76 and then we’ll just pretend that it’s the root node of a separate tree starting with 76 so

if there was anything below it then all those nodes would be included so this 76 right here it

really has nothing underneath it it’s a leaf which means well it can be the root node of its own

the subtree is just going to be a tree of one node.

So kind of boring, right?

You’re boring.

If instead we decided to look at the 33,

which is a little bit more interesting,

and we called the 33 the root node of its own subtree,

then really what we’re saying is all these nodes here

are included in that subtree.

So if I told you, give me the subtree starting with node 33,

then you would say, oh, it’s 33, 12, 39, 19.

descendants of the subtree root node that we picked out. So subtree just means, you know,

like a little fragment or a portion of the original tree. You could also say that the

entire tree is a subtree of itself. If you chose the subtree root to be the real root node, I mean,

that’s not super useful, but you can do it. Anyway, so if we decide to say that the 67 is the root of

with 67 and below in terms of descendancy is going to be considered part of the subtree.

I’ve highlighted the left subtree and the right subtree of the root node because that’s usually

what we say. We’ll say this is the left subtree over here and then over here we’re going to say

this is the right subtree. Meaning if you look at any node at all, if it has a left child,

then that left child is the root node of the left subtree of the node in question. Same thing for

same thing for the right so if I say all right let me duplicate this real fast

let me get rid of actually this real fast too if I say okay give me the left subtree of the 33 node

well then you would know to include the 12 and the 19 because the left subtree of the 30 node

has to the 33 node has to start with the left child of the 33 node which would be the 12 and

And we’ll just say, okay, the 12 is now the root node of its own subtree.

And then anything that goes below it in descendancy is going to be considered part of that subtree.

So that highlighted subtree is the left subtree of the 33 node.

The 39 is the right subtree of the 33 node.

I think I just did the wrong color.

Let me do that in gray.

Right, so we can do left subtree and right subtree for any node in the entire tree.

if a node has no children, then there are no subtrees, but we can still look and check.

And if there are children, then we’ve got subtrees or left and right subtrees.

Okay, so now that we’re done talking about subtrees real fast, let’s talk about the depth of a node.

So for me, I like to say that the depth of the root node is zero.

And so I’ll just, I guess we could start off by putting a zero on the 42 indicating it has zero depth.

It has zero depth.

Imagine maybe it’s a buoy in the water and it’s just like sitting, floating like directly

on the water.

So it has no depth.

It’s just like kind of on the surface.

But if you draw your binary search trees in this nice pretty way where every single time

you go down a generation from parent to child, from parent to child, you maintain, I guess,

like the same Y coordinate for same leveled nodes, then it’s really easy to calculate

the depth of every single node.

of every single note let me show you what show you what i mean real fast you saw my video uh

previously then you already know this but the 42 it’s got two children so if i go down to get one

of its children i’m going down to the 33 and then i’m going down to the 67 right since those two

children are on the same i guess level as if we were looking at a family tree they should be

physically on the same level they should be on the same y coordinate or the same horizontal plane

plane if we go down one more level which means any child of 33 or any child of 67 then those all

should be lined up also so notice how these are all lined up on the same y coordinate then if we

go down another level then this 19 here is just kind of by itself because the tree is not very big

so if we draw the tree like this which is a really smart way because uh it’s easier to debug

whether it’s a valid binary search tree and all sorts of other things,

then we can easily write down the depth kind of on the side of the graph.

We can say, all right, here’s depth zero, and here’s depth one,

and here’s depth two, and here’s depth three.

Just every time you go down one level, you just increase the depth.

And now you know the depths of all the nodes in the entire tree pretty quickly.

The 19, I’m just going to maybe do this in red.

the 56 and the 76 have a depth of 2, the 33 has a depth of 1, and the 42 has a depth of 0.

So all these trees, sorry, all these nodes in the tree have their own depth,

which are very easy to calculate if you draw the tree well. The next thing after depth is the height

of the tree. So what is the height of the tree? Well, that’s basically the depth of the deepest

to the very deepest node what is the minimum number of nodes that you must touch when you

start at the root node and then find your way to the deepest possible node in the entire tree so

um if you if you notice the 19 node is definitely the deepest node in the entire tree

it’s got a depth of three which means the height of the entire tree is four heights

equals four and maybe i’ll change that to like just black or something okay

So, let’s do it the other way real fast.

If we’re kind of just walking down the tree, let’s start at the 42 and then we go down

to the 33, we’ve touched two nodes so far.

We go down to the 12, we’ve touched three nodes.

We go down to the 19, we’ve touched four nodes.

So the height of the tree is four or the number of nodes that you need to touch as you make

your way down towards the deepest node or just a shortcut is the deepest nodes depth

plus one.

And that’s the height of the tree.

you can also have a height of a left subtree and a height of a right subtree so let me just

what’s going on here i think my thing is crashing hello oh i was definitely crashing i think my cpu

is burning right now all right i’m going to be complaining about my new cpu for a long time

i sprung only a few bucks for the best cpu that this motherboard could hold but it’s an old

something percent um i’m eventually going to have to like build a brand new computer

anyway so uh suppose we’re looking at the 67 node and the question is you know what is the height

of the left subtree of the 67 node versus the right subtree of the 67 node well if you recall

the left subtree is just all the nodes that are included uh beginning with the root node of the

subtree of the 67 node well that’s just the 56 node totally by itself what’s the right subtree

of the 67 it’s just that 76 node all by itself what’s the depth of 56 and 76 they’re both depth

zero if we’re talking about relative depths per their subtrees that means what is the height of

the left subtree and the right subtree they’re just one because the depth is zero the maximum

depth is zero and if we want to get to the deepest node in one of those subtrees we’re just

end up touching the one node that’s in the in the subtree at all so that means uh

uh i don’t want to write down uh left subtree height right now so left subtree height of 67

is one right subtree height of 67 is also one so we can do this with any node we want you know what

is the uh what is the situation with the 33 node let’s do uh the 33 node yeah uh it has a left

It has a left subtree height of 2 and you can tell because, well, the left subtree starts with the left child and the left child is going to be the root node of its own subtree.

Notice how the maximum depth we can find here is 1, right?

Like if we start at depth 0 for the 12, considering like it’s a relative depth, that means we take the deepest node, which is the 19 node, which has a depth of 1.

We add 1 to that.

So that means the height of that subtree is 2.

if you wanted to find the deepest node in that whole subtree, how many nodes would you have to

touch to get there? We’d have to touch the 12 and then touch the 19. We touch two nodes, so the height

of that subtree is two. Maybe I’ll just put H equals two here. So now for the right subtree of

the 33 node, it’s kind of easier. We just basically only have one node to really look at.

that means that 39 has a depth of 0 and the right subtree has a height of 1.

Whoops, 8 equals 1.

I didn’t put two equal signs. I like to do two.

I like to do the comparison operator.

We could also do the same thing with the 42 node, right?

We can say, let’s get rid of all this stuff real fast.

We can do the 42.

Its left subtree starts with that 33 node.

33 node so I’m just going to highlight that real fast the 42 nodes left subtree has a height of

one two three and you can tell because the 33 has a depth of zero a relative depth of zero and

the 12 and the 39 have one and the 19 has two so the deepest node has a depth of two

so that means the height of that subtree is is three so I’m just going to like do this real fast

and then the right subtree of the 42 node, the root node of the entire tree, is going to be this.

So the root node has a depth of zero, depth of zero, and then these other leaves over here have

depth of ones, which means the height of this subtree is going to be two, or the deepest node

plus one, or the number of nodes you need to touch to find the deepest node. And 56 and 76,

those are both equally the deepest node in those trees okay so we talked about a bunch of

terminology here let me just double check my notes in case I forgot to to tell you anything

I think I’m all right well maybe okay maybe I should real fast just briefly mention that these

nodes I mean this is not really part of the video exactly but let’s let me just mention that these

whoops that’s dumb let me do a blue circle they would have you know there’s

like some sort of an object you would call it a node and then they would have

pointers they would have each of these nodes would have a pointer to its

parents and it would have a left child pointer that goes down into the left and

a right child pointer that goes down to the right and also a little slot in that

object for the data so I’m just going to put t type data and C++ I usually say

I usually say that the templated data type for a node or data structure is just the T type.

That just means you could put anything you want.

You could have your nodes hold integers, letters, strings, custom objects, whatever you want to do.

And we’ll talk about this more in future videos.

But long story short, I want you to just imagine that every node actually has three pointers inside of it pointing to something else, to other nodes,

graph because in a graph you could have like a whole bunch of different connections and that’s

usually managed by the actual graph object itself so anyway we’re done with terminology i hope you

learned a little bit of stuff and had a little bit of fun thanks for watching this video

tell your friends eat a donut and a chocolate and then be really happy and stuff okay i gotta go

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The post BST Terminology: Root, Leaves, Subtrees, Depth, Height, Ancestors & More appeared first on NeuralLantern.com.

Clear step-by-step explanation of what defines a Binary Search Tree (BST). We build the definition rule-by-rule starting from graphs all the way to the BST ordering property (all left descendants, node, all right descendants). Great first video before learning insert, delete, search, and Big-O.

What is a Binary Search Tree? 00:00
Intro to Graphs and Nodes 00:42
Connected Graph Requirement 03:16
Acyclic Graph – Removing Cycles 04:54
Turning Graph into Tree 07:05
Establishing a Root 07:09
Adding Hierarchy and Levels 09:05
Single Common Ancestor 11:18
Binary Tree Definition 17:30
BST Ordering Property 17:39
Left Subtree Less Than Node 18:29
Fixing Invalid BST Example 20:26
Valid BST Final Check 21:24
In-Order Ascending Order 24:11
Video Summary and Next Steps 25:02
Thanks and Call to Action 25:30

=-=-=-=-=-=-=-=-=

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hey there let’s talk about binary search trees specifically in this video I want to help you

define and identify what is a binary search tree all the rules that go into determining whether

or not something is a binary search tree so you can you can tell that you’re looking at one or

whether you’re not looking at one we’ll talk about some terminology but keep in mind I’m

going to save searching and sorting and like you know deleting and adding items and a whole bunch

extra stuff, especially the hard stuff for videos that’ll come right after this one.

For now, we’re just going to talk about what is a binary search tree.

So you can see on the screen right here, I hope that we have a binary search tree.

Let’s see.

Can you see that?

Yeah, I think you probably can.

Yeah, I think you can.

This is not drawn quite as well as I would like to draw ours.

I’ll tell you how to do that in a little while.

to do that in a little while but when you draw a tree in a really really nice pretty aligned way

it’s way easier to debug it’s way easier to tell that you’re looking at something that is sorted

and so forth anyway okay so first let’s start off with a graph so this is not a video about graphs

uh i guess i can probably expect that you already kind of know what a graph is at this point in time

and edges. So what do I mean by nodes? Well just imagine there’s like a little circle here

that contains some sort of data. If you know data structures already then

there would be a t-type sitting inside the node. We’ll just call the t-type an integer

and so for our nodes, for our trees, we’re just going to say that they hold integers even though

you know technically speaking in a binary search tree or a graph you could have all sorts of

Okay. So I’m just going to, you know, grab this and sort of like duplicate it and just duplicate it again.

I’m just going to make a bunch of duplicates. Something is definitely wonky about my setup right now.

I think I just upgraded the camera and the CPU is maxed out. I just upgraded the CPU on this computer too.

Oh well. Okay. So we aren’t going to support duplicate values in our binary search tree,

I’m going to say it doesn’t really matter for the graph. Later on, we’ll start making sure there’s

no duplicate data. For now, I think I can probably do that somewhat quickly. Let’s see. So a graph,

you know, for the for our purposes right now, a graph is just basically a collection of nodes and

edges. It could be empty. It could be just nodes. It couldn’t just be edges, it would have an edge

always has to have a node or two connected to it. So we just have like a bunch of nodes inside of

data values this is a valid graph but I’m going to draw some edges randomly here so we’re going

to do this it doesn’t really matter I’m just randomly drawing edges okay actually I should

probably do maybe like one that’s kind of separated okay so this is a graph this is not a binary search

tree so let’s add a bunch of rules let me just pull up my rules real fast here okay so what is

First we’ll start off with the graph like we have here and then we’ll say that the graph must be connected.

What is a connected graph? A connected graph is basically a graph where every single node can find every single other node

or find a path to every single other node only following along edges and only by respecting the direction of edges.

You can see this particular graph is undirected meaning the edges don’t really have a direction to them.

So that means we could travel along in any direction from node to node.

So just as a real quick, you know, I guess like tutorial, I’m going to say is 8 connected to every single other node?

Well, it can find a path to 12 and it can find a path to 33 and it can find a path to 76 and 45.

But it cannot find a path to 99, which means this is not a connected graph.

So that’s the first rule that we have to implement here.

We’re going to say this graph needs to be upgraded or modified so that it will be a connected graph.

start randomly adding edges until this is a connected graph I’m gonna do maybe

like an edge from 99 to 12 there and then now I think this is a connected

graph but let’s double check real fast so connected graph you know what I’m

gonna put graph connectedness in another video to keep this one short for now

check out my other videos if you would like to see whether or not a graph is

but long story short as I just said if every single node can find a path to

every single other node without you know skipping where there is no edge then

it’s connected okay so that’s the first thing the next thing that we need let me

just put the words connected up here we need a connected graph the next thing

that we need is we need an acyclic graph which basically means a graph with no

A cycle is, can you find a path in this graph that starts at one node and ends at the same node without repeating any edges?

So for example, if I was asking, you know, is 76 involved in any cycles?

Well, we could go from 76 to 33 and then 12.

And, you know, we could kind of take 8, 45, 12 again.

But by the time we turn around and try to get back to the 33, we’ve already crossed this edge twice.

so that means the 76 is not involved in any cycles because we cannot find our way back to 76

without repeating an edge so 76 is okay i’m just going to maybe put like a little check box here

and uh you can also tell that the 33 is not involved in any cycles because we go down to

the 76 and then come back up to 33 we already repeated an edge so that’s not valid same thing

again. Same thing for the 34, same thing for the 99. But if you look at the 12 here, we could

actually find a cycle. We could go from the 12 to the 8, and then from the 8 to the 45, and from the

45 to the 12. Now we’re back where we started. We’re at the 12 node again, and we did not repeat

any edges. So this is not a cyclic. This is a cyclic graph. It’s got a cycle. So that’s the

we want to make sure that our graph has no cycles. So, uh, the next rule I’m just going to say is,

is going to be satisfied by me removing some edges at random. So I don’t know, let’s get rid of, uh,

the eight to the 45 connection. See if that works. Um, let’s see. Well, wait a minute. What did I

just do? Something happened here. I just accidentally, oh, I erased too many edges

from the last slide. Okay. Let me add that one back in 33 to 76. That was supposed to be there.

33 to 76 that was supposed to be there.

So now we have an acyclic graph

and we can consider this graph a tree.

The next thing we need to do is add a hierarchy to this tree.

So this is gonna make more sense in a little bit

but for now I’m gonna say,

we’ll imagine a family tree kind of,

it’s not really exactly a family tree

but you know imagine there are parents

and children and grandchildren

and we’ll just try to like,

we’ll try to rank children according to

according to you know when they were born or or how many parents or children

they have you know their descendancy their lineage whatever you want to call

it so I think I’m just gonna maybe move the eight up a little bit so that it’s

on the same rank with the 12 and the 33 so I’m just gonna move these up a little

bit and then redo their connecting lines so I’m gonna do this and that and then

and then we’ll say that the 76 is still a child of the 33.

So I’ll just kind of move it over here.

Or sorry, we never said it was a child.

Now we’re saying it’s a child

because when you go from top to bottom,

imagine like a family tree,

we’ll say we have parents and children.

The children or the descendants are lower.

The ancestors or the parents or the grandparents are higher.

Maybe I want to put this 45 over on the left

so that it’s a child of the 8 node.

And maybe I’ll say that the 99 node

the 99 node is a child of the 12 node. Okay, so eventually this is going to look not exactly like

a family tree because usually in the family tree we have two parents that go to one or more children.

In this type of tree, in eventually our binary tree, we’re going to have one parent that has,

you know, children just by itself, like asexual reproduction. We’re pretending to be bacteria

What was that comedy skit a long time ago, Tiny Elvis?

And he was like, hey, see that protozoa over there?

The thing is huge.

Anyone?

No?

Okay.

So the next rule we need is we must have a rooted tree.

What is a rooted tree?

Actually, let me clean up the lineage real fast, the heritage.

I want to make nodes that have the same level with respect to their ancestors

respect to their ancestors to have the same physical y coordinate the same

physical level so notice how the 99 I just moved it down a little bit because

I wanted it to look like it’s the same you know number of generations lower

than the top row so like with aid it had a child 45 so that’s one level down 33

had a child that’s one level down that’s a 76 and the 12 also one level down

would have been 99 but the 99 was kind of too physically high this will help

will help you debug your binary trees in the future anyway so i’m just going to clean this up right now

and uh now that we have you know parent child relationships we need to look at the most common

ancestor in the whole entire tree or if there is one so notice how uh i guess i’m just kind of using

the word ancestor without explaining it so your ancestors are the people that you’re disappointing

or whatever it is, ancestors came before us, right?

Like my parents are my ancestors.

My grandparents are also my ancestors.

My great-grandparents are also my ancestors.

Going down in the family tree are your descendants.

You know, my children are my descendants.

My grandchildren are my descendants.

My great-grandchildren are my descendants and so forth.

So if you look at the number 34 here,

what is the greatest ancestor that it has?

Well, if you just go up and up and up

until you can’t go up anymore,

up until you can’t go up anymore it’s the 12th so that means the greatest ancestor of at least in

this tree of the 34 is the 12th the 45 the highest you can go is the 8 notice how 8 and 12 and 33 are

siblings there’s not really like a parent-child relationship they’re connected sideways we’ll

eventually get rid of the sideways connections for our binary search trees but for now we have

some siblings and so we actually have three different greatest ancestors in this tree we’ve

tree we’ve got 8 and 12 and 33 and that’s no good so the next rule that we need is we need

one common ancestor for every single other node in the entire tree so I’m going to duplicate this

slide right here and I’m going to say we need to clean this up so we got to choose 8 or 12 or 33

I don’t know I’m just going to maybe say that 8 and it’s 45 are children of the 12 so I’m just

line here arbitrarily so i’m gonna go like that notice how i’m trying to keep the 8 the 99 and

the 76 on the same y coordinate the same level now we have two uh you know greatest ancestors

we need to have only one so i think i’m probably going to say that the 33 and the 76 are going to

become children of the 12 node just for simplicity so i’m going to put this over here i’m going to

i’m gonna like move this over here this is a gross tree but we’ll fix it soon we’ll do that all right

okay okay

okay and then i have to get rid of that little sibling line and then i have to add another line

to say that 33 is a child of 12. okay so now we have a rooted tree what we had before was an

unrooted tree or just like not a rooted tree because there was no root there was no greatest

greatest common ancestor to the entire rest of the tree. But now we can say that 12 is the root

of the tree because it’s the ancestor for all other nodes. I guess even if 12 was by itself,

we could call it a root of tree. It just wouldn’t be a very good example. So I’m going to actually

get rid of the word acyclic here since we already handled that. And I’m going to do,

I’m going to say that 34’s greatest ancestor is 12, 99’s greatest ancestor is 12, 33’s greatest

all the nodes in the tree their greatest ancestor is 12 and it’s common to all of them so now we

have a rooted tree maybe I should type in my doomed I really hope the screen I really hope

the screen cast doesn’t freeze it probably will upgraded the CPU it’s 33 faster than it was before

board so i am still using an old cpu even though it’s new to me dang it i’m gonna have to spend

like 500 okay uh so the next rule is so we have a rooted tree and now let’s rearrange everything

hierarchically which we already kind of did so uh there’s not really much to do here except

we did hierarchical organization

we probably just need to move on to our next rule which is to say that

if our rooted tree is to become a binary tree

then each node in the entire tree

cannot have more than two children

every single node in the entire tree

can have zero or one or two children

if there are three or more anywhere in the entire tree

then it’s not a binary tree

so okay that means you know the 45 is okay

because it has no children

because it has no children. All the rows, all the leaves at the bottom are okay. All these nodes

over here are okay because they just have one child. But the 12 node, the root of the tree,

it’s got three children and that violates the rule. So now we need to rearrange this.

So I think I’m going to just move these numbers down here maybe. Whoops. Maybe move these nodes

descendants of the 33 node. So I’m going to do this.

Okay. So cool. Now we have a binary tree and now we need to rearrange the drawing a little bit.

This is not part of the definition, but this will make it easier to debug your trees. So what I want

to do is make it so that every left descendant of a node appears physically on that node’s left

every right descendant of a node should appear on its right side

or like further to the right on the x-coordinate

so sort of similar to you know like the y-coordinate ranking

but notice how the 12 its left child is 8 and its right child is 33

so that means all the nodes here on the left side you know that come underneath the 8

are left descendants or you could say that they are in the left subtree of the 12.

So, they’re okay because 8 is to the left of 12 and 45 is to the left of 12.

But notice how 33 is a right child of 12 and it’s part of the right subtree.

It’s part of the right descendancy, but it’s physically on the left of 12.

That’s going to be really bad because it’ll be harder to debug our trees later.

So, I’m just going to slide it over without rearranging the tree.

And I usually try to split the difference here.

and then their parent is kind of like physically in the middle that will also help you debug in

the future so i’m going to do this wrong let me get the blue i’m going to do this make sure that

my ranking is still okay not the best most neatest tree and so now i’m going to look at the eight

it’s left subtree or it’s left uh descendancy uh all of those nodes which is just the 45 are

it’s left child 76 is the only thing that’s on the left side in the left subtree so that’s all

right probably the 33 I think it’s still a little lopsided I probably could have you know dragged

it a little bit more to the right but I’m going to leave it so the 33 and the 76 are okay and then

the 99 is a right descendant or it’s part of the right subtree of the 33 so it should be on the

right side it is that’s good the 34 should be on the right side of the 33 because it’s it’s a it’s

It’s part of the right subtree of 33.

It’s part of the right descendancy.

And it should also be on the left side of the 99

because it’s part of the left subtree of the 99.

So, so far, this is okay.

This is not the best drawing.

So now we have a binary tree

and we have kind of drawn it in a slightly good way.

Not super great.

The last rule that we’re going to add here is,

whoops, what happened there?

I’m supposed to do that, yeah.

The last rule we’re going to add is

we must have ordering to our nodes, to the numbers in our nodes. So what do I mean by ordering? Well,

the whole reason that we want a binary search tree, at least usually, is because it’s really,

really fast to search through. This is going to be a tree that we can search through in log time,

which we’ll talk more about in another video. We’ll do time complexities and searching and stuff. But

in order for this to be a super fast tree to search through, we need to be able to

very quickly make a decision. Do we go down and to the left? Do we go down and to the right?

left do we go down into the right and every decision we make should help us eliminate half

of the remaining data set half of the tree at a time every time we go down a level and that’s only

possible if the nodes themselves are ordered so what i’m going to do is say every node that’s on

the left side or let’s say every descendant that’s on the left side of a node should be of a lesser

value than that particular node every node that is on the right side of a node meaning it’s a right

descendant of that node or part of the right subtree of the node should be greater than that

node. Please note that this would not allow us to support duplicate values in our tree.

You could use something like less than or equal to versus greater than if you wanted to support

duplicate values in your tree. You could also change the T type, you know, the template type

to a node from the data you want it to store to a list. So every single node has a list inside of it

and the items in the list are just, you know, the regular values that you wanted. There’s a whole

values that you wanted there’s a whole bunch of different ways you could actually do that

but for now we’re going to see our trees don’t support duplicate values

so okay we don’t support duplicate values but we need you know from left to right

the order needs to look sane and this is kind of why we’re drawing the tree in this special way

we’re trying to make our tree really easy to debug visually so check this out the way we’ve

drawn this every single node has its own x coordinate basically notice how the 12 there’s

nothing underneath it at least not directly underneath it the 33 the 34 all the nodes you

know you can draw a straight line down from the node and it shouldn’t cross over with any other

nodes or edges so that means now that we’ve drawn it this way and we’ve made sure that all the left

descendants are on the left and all the right descendants are on the right we could actually

just scan our eyes from left to right and be sure that we are seeing an increasing order or

Since duplicates aren’t allowed in this tree, we’ll say an increasing order or an ascending order.

So if we start at 45 and then we go to the right, we should see a higher number.

So if we look at 8, 8 is a lower number.

So this violates the ordering rule.

This is not a valid binary search tree.

It’s only a valid binary tree.

So sad.

Okay, so I could, you know, swap the 45 and the 8 or I could just increase the value of the 8 itself.

of the 8 itself, I think it’ll be faster for me to just increase the 8 value. So I’m going to make

it 52 now. So we’ve got 45, 52. Then we go to the right a little bit more. It should also increase.

So 52 should increase. I’m just going to randomly increase it to, you know, 59, let’s say. Then as

we go to the right, we see another number here. It is 76. That already is increased from 59.

So we don’t need, we don’t need to alter that number. Then we go to the right one more time

that because it’s lower than 76 so I’m going to put 87 then when we go to the right one more time

that’s the 34 down here that’s definitely a decrease so we have to increase it now so I’m

going to say 99 how about if I just do 92 yeah so now we have a 92 down here and then if we go

to the right a little bit more there’s the 99 and now we actually have a binary search tree let’s

just double check all the rules first off is this a graph yeah there’s nodes and edges there’s nothing

There’s nodes and edges.

There’s nothing weird happening.

There’s not like a edge floating in the middle of nowhere.

Is this a connected graph?

Yeah, it is.

Because every single node, if you followed, you know, the edges, you know, could find

every single other node.

Like the 45, it could find, you know, the 92 node if it just followed all these edges.

Notice again that these edges have no direction.

There’s no arrows on them.

So we could go up or down in any direction we want.

So that means the 76 can find everything else.

else and so this is a connected graph for sure okay now that we have a

connected graph is this an acyclic graph I don’t know about you but I can’t find

any cycles in the graph right like the 92 it goes out but it doesn’t come back in

same thing for the 76 if we went from the 76 up to the 87 and then took a

right turn we wouldn’t be able to come back towards it without repeating that

same edge so this is definitely an acyclic graph all of the nodes follow

they’re not involved in cycles is this a rooted tree yeah there’s only one common ancestor to the

entire tree it’s the 59 it’s the highest node in the entire tree and it’s common to all other of

all of its descendants um so let’s see i’m going to duplicate this real fast the next thing is

we arranged it hierarchically and we added the terminology of like i think i kind of just like

tried to sneak this one past you but we added the terminology of a left child and a right child

and a right child.

Right? We also added the terminology of a right subtree and a left subtree.

Just to clarify what I mean, what do I mean by a right subtree and a left subtree.

So imagine we’re considering the 59.

If we consider the 59, then the 52 is the root node of the left subtree of the 59 node.

If we consider the 59, then the 87 is the root node of the right subtree of the 59 node.

so the right child is basically the root of the subtree uh in in you know the right child is the

root of a subtree of the right subtree uh i think i’m getting muddled my words here okay

anyway so we have a little bit of terminology uh the 59 is the root node that’s like the

the only node in the entire tree without uh an ancestor or a parent so okay uh then we have a

Note, and the entire tree doesn’t have more than two children.

Notice how the 59 has two children.

That’s fine.

If it was three, that’s way too much.

52 has one.

87 has one.

No, 87 has two children.

99 has one child.

92 has zero children.

So, okay, we’re not violating that rule.

So, we definitely have a binary tree.

And then we, you know, kind of read from left to right.

Let me write down the numbers this time for fun.

physically without kind of like following edges or anything we just scan

our eyes from left to right we can see that we have an ascending list if we

supported duplicates it would be okay to just see a non decreasing list but we

see an increasing list or an ascended list ascending list okay whoops this is

left to right so this is actually a binary search tree and um i don’t know about you but i think i

need to like you know get a snickers or something right now because uh this video has been going on

for longer than i wanted it to go on so we’ve defined a binary search tree in future videos

we’ll talk about how to add nodes into a binary search tree how to build the tree what’s the time

complexity of the tree in various operations deleting nodes from a tree terminology like

terminology like what are these nodes called things like that so anyway thank you so much

for for watching i hope you learned a little bit of stuff and had a little bit of fun

i’ll see you at a later date in time

just kidding

hey everybody thanks for watching this video again from the bottom of my heart i really

appreciate it i do hope you did learn something and have some fun uh if you could do me a please

If you could do me a please a small little favor

Could you please subscribe and follow this channel or these videos or whatever it is you do on the current social media?

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So please do do me a kindness and uh and subscribe

I’m sleeping in the middle of the night and I just wake up because I know somebody subscribed or followed.

It just wakes me up and I get filled with joy.

That’s exactly what happens every single time.

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